# Volume Approximation of Strongly ${\mathbb{C}}$-Convex Domains by Random
Polyhedra

Siva Athreya  Theoretical Statistics and Mathematics Unit, Indian Statistical
Institute, Bangalore<EMAIL_ADDRESS>,  Purvi Gupta  Department of
Mathematics, Indian Institute of Science, Bangalore<EMAIL_ADDRESS>and
D. Yogeshwaran  Theoretical Statistics and Mathematics Unit, Indian
Statistical Institute, Bangalore<EMAIL_ADDRESS>
###### Abstract.

Polyhedral-type approximations of convex-like domains in ${\mathbb{C}}^{d}$
have been considered recently by the second author. In particular, the decay
rate of the error in optimal volume approximation as a function of the number
of facets has been obtained. In this article, we take these studies further by
investigating polyhedra constructed using random points (Poisson or binomial
process) on the boundary of a strongly ${\mathbb{C}}$-convex domain. We
determine the rate of error in volume approximation of the domain by random
polyhedra, and conjecture the precise value of the minimal limiting constant.
Analogous to the real case, the exponent appearing in the error rate of random
volume approximation coincides with that of optimal volume approximation, and
can be interpreted in terms of the Hausdorff dimension of a naturally-
occurring metric space. Moreover, the limiting constant is conjectured to
depend on the Möbius-Fefferman measure, which is a complex analogue of the
Blaschke surface area measure. Finally, we also prove $L^{1}$-convergence,
variance bounds, and normal approximation.

###### Key words and phrases:

Strong ${\mathbb{C}}$-convexity, random polyhedra, Poisson process, binomial
process, volume approximation, optimal approximation, limit theorems, normal
approximation.

###### 2020 Mathematics Subject Classification:

60D05, 32F17, 60G55, 60F05

###### Contents

  1. 1 Introduction
    1. 1.1 Context and Contribution
  2. 2 Limit theorems for random polyhedra
    1. 2.1 Polyhedra induced by strongly ${\mathbb{C}}$-convex domains
    2. 2.2 Main results
    3. 2.3 Remarks
  3. 3 Geometric estimates
    1. 3.1 Preliminaries
    2. 3.2 Depth estimates
    3. 3.3 Area and volume estimates
    4. 3.4 Containment and coverage
    5. 3.5 Nearest-neighbor estimates
  4. 4 Proofs of limit theorems
    1. 4.1 Theorem 2.3: convergence in probability
    2. 4.2 Theorem 2.4: $L^{1}$-convergence and variance bounds
    3. 4.3 Theorem 2.5: central limit theorem
    4. 4.4 Proposition 2.6: conditions for coverage
  5. A Background on Poisson and Binomial processes
  6. B The Möbius geometry of strongly ${\mathbb{C}}$-convex domains
  7. C Area and volumes estimates for model domains

##  1\. Introduction

Approximation of convex bodies by (convex) polyhedra is an important topic in
convex geometry; see Gruber [13] and Bronstein [7]. Motivated by both
theoretical and practical considerations, this has led to numerous studies of
approximation of convex bodies by random polyhedra; see Reitzner [33] and Hug
[18]. A fascinating feature of such studies is that for both deterministic and
random approximations, the asymptotic rates and leading constants are rather
well understood. For instance, the least possible volume discrepancy between a
convex body $K\subset\mathbb{R}^{d}$ and the convex hull of $n$ points on its
boundary decays like $v^{\sharp}_{\mathbb{R}}(K)n^{-2/(d-1)}$, where the
limiting constant $v^{\sharp}_{\mathbb{R}}(K)$ is given in terms of the
Blaschke affine surface area measure of $K$. This was shown by McClure and
Vitale [28] for $d=2$, and by Gruber [14] for all $d$. On the other hand, the
least possible volume discrepancy between $K$ and the convex hull of $n$
i.i.d. random points chosen according to a continuous density on its boundary
decays like $v^{*}_{\mathbb{R}}(K)n^{-2/(d-1)}$, where $v^{*}_{\mathbb{R}}(K)$
differs from $v^{\sharp}_{\mathbb{R}}(K)$ only by a dimensional constant.
Here, “least” is to be understood as over all possible continuous densities on
the bounday. This was proven by Schneider [38] in the case of $d=2$, Müller
[29] for balls in all dimensions, and by Schütt and Werner [39] and Reitzner
[31] for general convex domains (under different regularity assumptions).
Similar behaviour has also been observed for circumscribed polytopes in [14]
and random circumscribed polytopes in Böröczky Jr. and Reitzner [6].

While there are several notions of convexity in several complex variables, the
asymptotic analysis of polyhedral approximations in the complex setting is
rather underexplored. Deterministic volume approximations for two different
classes of (convex-like) domains in $\mathbb{C}^{d}$ were investigated by the
second author in [15] and [16]. In both cases, it was either shown or
conjectured that for a domain $D\subset\mathbb{C}^{d}$, the error in volume
approximation decays like $v^{\sharp}_{\mathbb{C}}(D)n^{-1/d}$, where
$v^{\sharp}_{\mathbb{C}}(D)$ depends on complex analogues of the Blaschke
surface area, i.e., the Fefferman hypersurface measure in the former case, and
the Möbius–Fefferman hypersurface measure in the latter case. In this article,
we initiate the study of random polyhedral approximations in the complex
setting by exploring random analogues of the approximation considered in [16].
We determine the rate of approximation, and make a conjecture for the limiting
constant; see Page 7 for an an overview of the results.

#### Organization of the paper.

For the convenience of the reader, we summarize the central objects and the
main results of this paper in Section 1.1. We also provide some motivation and
context, as well as describe the key proof ideas in this section. Section 2
expands on Section 1.1 and contains the statements of the main results of this
paper. The underlying domains and polyhedral sets are defined in detail in
Section 2.1. The main results are stated in Section 2.2. We end Section 2 with
some more remarks on the main theorems (Section 2.3). The proofs of the main
results are spread over Sections 3 and 4. In the former, we collect some
recurring notation and all the geometric (deterministic) ingredients needed
for the proofs. Finally, the main (probabilistic) proofs are in Section 4,
where each subsection is devoted to the proof of each main result. In Appendix
A, we recall some well-known results on Poisson and binomial processes that
are used in our proofs. In Appendices B and C, we collect all the background
material on the geometry of ${\mathbb{C}}$-convex domains that plays a role in
Section 3. In an attempt to make the article accessible to both probabilists
as well as complex analysts, we have provided more background details than is
customary in either of the two fields.

#### Convention on constants.

Throughout the paper, $c$ and $C$ will denote generic positive constants whose
values may change from line to line. The dependence of these constants on
specific parameters of the problem will be indicated via subscripts as in
$C_{g,D},C_{D}$, etc.

###  1.1. Context and Contribution

Convex geometry has had fruitful interactions separately with probability and
several complex variables. We envisage that the joint interaction between the
three subjects could be equally exciting. Apart from mere mathematical
curiosity, the motivation behind considering ${\mathbb{C}}$-convexity is to
enlarge the class of convex-like domains for which approximation problems can
be studied. In geometric terms, ${\mathbb{C}}$-convexity is the natural
analogue of classical convexity in complex projective geometry. Although there
is a developing literature on random polytopes in non-Euclidean geometry (see,
for example, Besau and Thäle [3], Besau et al. [4]), our work is fundamentally
different in that the underlying notion of convexity itself is non-Euclidean
(complex projective).

#### Strongly ${\mathbb{C}}$-convex domains.

Recall that for a $\mathcal{C}^{2}$ domain $K\subset\mathbb{R}^{d}$, convexity
and strong convexity are characterized by the semi-positive definiteness and
positive definiteness, respectively, of the second fundamental form of $bK$,
the boundary of $K$ (viewed as a smooth manifold), at all points of $bK$.
Analogous to this, a $\mathcal{C}^{2}$ domain $D\subset\mathbb{C}^{d}$ is
${\mathbb{C}}$-convex (strongly ${\mathbb{C}}$-convex), if for all $w\in bD$,
the restriction of the second fundamental form of $bD$ to $\mathcal{H}_{w}bD$
is semi-positive definite (positive definite), where $\mathcal{H}_{w}bD$ is
the maximal complex subspace of the real tangent space of $bD$ at $w$ when
viewed as a subspace of $\mathbb{C}^{d}$. We refer to $\mathcal{H}_{w}bD$ as
the complex tangent space to $bD$ at $w$. In this paper, we work with bounded
strongly ${\mathbb{C}}$-convex domains in $\mathbb{C}^{d}$. However, for the
sake of convenience and illustration, the examples given in Figures 1, 3 and 4
are of unbounded strongly ${\mathbb{C}}$-convex domains. The planar case
($d=1$) appears to be simpler since complex tangent spaces reduce to points,
so there is no complex geometry to speak of; compare Figure 2 ($d=1$) and
Figures 3 and 4 ($d=2$). Thus, the case $d=1$ calls for a separate
investigation, and we exclusively deal with the case $d>1$ in this paper.

\begin{overpic}[grid=false,tics=10,scale={0.5}]{drawing.pdf} \put(240.0,90.0){
$|z_{1}|$} \put(5.0,80.0){ $x_{2}$} \put(83.0,172.0){$y_{2}$}
\put(200.0,58.0){$bD$} \end{overpic} Figure 1. A non-convex strongly
${\mathbb{C}}$-convex (unbounded) domain in ${\mathbb{C}}^{2}$:
$D=\\{y_{2}>|z_{1}|^{2}-x_{2}^{2}\\}$.

We postpone a broader discussion on ${\mathbb{C}}$-convexity to Section 2.1,
but note that (real) convex sets in $\mathbb{C}^{d}\cong\mathbb{R}^{2d}$ are
always ${\mathbb{C}}$-convex but the converse is not true; see Figure 1. While
real convexity (and the class of real convex polyhedra) is invariant under
real affine transformations, ${\mathbb{C}}$-convexity is invariant under the
(larger) group of complex projective transformations, making it a suitable
notion of convexity for complex projective geometry. From this point of view,
we shortly introduce a class of polyhedral sets that is invariant under the
intermediate group of complex affine transformations. In complex analysis,
${\mathbb{C}}$-convexity plays a significant role in questions of duality,
interpolation, integral representations of holomorphic functions and metric
geometry; see Andersson et al. [1], Barrett [2], Lanzani and Stein [22] and
Lempert [25], for instance.

#### Induced polyhedra.

In the real setting, given a finite set $\varphi=\\{w^{1},...,w^{n}\\}$ on the
boundary of a real convex domain $K\subset\mathbb{R}^{d}$, the simplest
construction of a convex polyhedron is to take the convex hull
$P_{\text{i}}(\varphi)$ of the point set $\varphi$, i.e.,
$P_{\text{i}}(\varphi)$ is the intersection of all the convex sets containing
$\varphi$. In the complex setting, since the intersection of two
${\mathbb{C}}$-convex sets need not be ${\mathbb{C}}$-convex, there is no
notion of a “${\mathbb{C}}$-convex hull”; see [1, Section 2.2]. A more natural
construction of polyhedra is suggested by the following construction of
circumscribed polyhedra in the real setting. Given a finite set
$\varphi=\\{w^{1},...,w^{n}\\}$ on the boundary of a strongly convex domain
$K\subset\mathbb{R}^{d}$, let $P_{\text{c}}(\varphi)$ denote the the
intersection of those $n$ half-spaces determined by the tangent planes to $bK$
at $w^{1},...,w^{n}$ that contain $K$. Analogous to this construction, a class
of “complex” polyhedra are defined in [16] as follows. Given a strongly
${\mathbb{C}}$-convex domain $D\subset\mathbb{C}^{d}$, a source set
$\varphi=\\{w^{1},...,w^{n}\\}\subset bD$ and a size function
$\boldsymbol{\delta}:bD\rightarrow(0,\infty)$, let
$P(\varphi;\boldsymbol{\delta})$ be the union of those connected components of

(1.1)
$\bigcap_{j=1}^{n}\left\\{z\in\mathbb{C}^{d}:\text{dist}(z,\mathcal{H}_{w^{j}}bD)>\boldsymbol{\delta}(w^{j})\right\\},$

that intersect $D$, where $\mathcal{H}_{w}bD$ is the complex tangent space to
$bD$ at $w$; see Section 2.1. The set (1.1) is ${\mathbb{C}}$-linearly convex
and hence, $P(\varphi;\boldsymbol{\delta})$ is weakly ${\mathbb{C}}$-linearly
convex; see [1, Section 2.1]. We call $P(\varphi;\boldsymbol{\delta})$ an
induced polyhedron with at most $n$ facets. Though “$D$-induced polyhedra” may
be a more precise term, we drop the domain-dependence for convenience. In
comparison, $P_{\text{i}}(\varphi)$ and $P_{\text{c}}(\varphi)$ have at most
$n$ vertices and at most $n$ facets, respectively. Geometrically, the role of
half-spaces is played by complements of ‘tubes’ of radii
$\boldsymbol{\delta}(\boldsymbol{\cdot})$ with central axes
$\mathcal{H}_{\boldsymbol{\cdot}}bD$. See Figure 2 for illustrations of
induced polyhedra in dimension one. In higher dimensions, induced polyhedra
are hard to draw, but see Figures 3 and 4 for illustrations of these ‘tubes’
that generate the polyhedra.

\begin{overpic}[grid=false,tics=10,scale={0.45}]{drawing-2.pdf}
\put(65.0,-15.0){$(a)$} \put(255.0,-15.0){$(b)$}
\put(100.0,20.0){\small{\color[rgb]{0.68,0,0}$w$}}
\put(108.0,4.0){{\color[rgb]{0,.5,.5}\footnotesize$\\{z:\text{dist}(z,\mathcal{H}_{w}(bD))=\delta\\}$}}
\end{overpic}
Figure 2. Induced polyhedra in ${\mathbb{C}}$: the domain $D$ is the region
bound by the black curve; the induced polyhedron is the union of regions
shaded in pink.
$(a)$ A disconnected and non-contained induced polyhedron with $21$ facets.
$(b)$ A connected and contained induced polyhedron with $52$ facets.
\begin{overpic}[grid=false,tics=10,scale={0.5}]{drawing-4.pdf}
\put(240.0,95.0){ $|z_{1}|$} \put(240.0,115.0){ \small$\mathcal{H}_{w}(bD)$}
\put(66.0,108.0){\small{\color[rgb]{0.68,0,0}$w$}} \put(5.0,80.0){ $x_{2}$}
\put(83.0,172.0){$y_{2}$}
\put(-60.0,140.0){\footnotesize{\color[rgb]{0,.5,.5}$\\{z\in\overline{D}:\text{dist}(z,\mathcal{H}_{w}(bD))=\delta\\}$}}
\put(200.0,58.0){$bD$} \end{overpic} Figure 3. A “cut” $C(w;\delta)=\\{z\in
D:\text{dist}(z,\mathcal{H}_{w}(bD)<\delta)\\}$ (region bound by the teal tube
and $bD$) and a “cap” $S(w;\delta)=\\{z\in
bD:\text{dist}(z,\mathcal{H}_{w}(bD)\leq\delta)\\}$ (in yellow) in the domain
$D=\\{y_{2}>|z_{1}|^{2}-x_{2}^{2}\\}$ at $w=(0,0)$. The interior of the
complement of $C(w;\delta)$ in ${\mathbb{C}}^{2}$ is an induced polyhedron
with one facet.
\begin{overpic}[grid=false,tics=10,scale={0.45}]{drawing-1.pdf}
\put(70.0,240.0){$y_{2}$} \put(130.0,240.0){$bD$} \put(15.0,115.0){ $x_{1}$}
\put(180.0,140.0){ $y_{1}$} \put(100.0,120.0){ $\mathcal{H}_{w}(bD)$}
\put(82.0,142.0){{\color[rgb]{0.68,0,0}$w$}} \put(70.0,100.0){ $(a)$}
\put(15.0,160.0){\footnotesize{\color[rgb]{0,.5,.5}$\delta$}}
\put(323.0,240.0){$y_{2}$} \put(400.0,240.0){$bD$} \put(330.0,100.0){ $(b)$}
\put(260.0,115.0){ $x_{2}$} \put(450.0,140.0){ $y_{1}$} \put(425.0,160.0){
\small$\mathcal{H}_{w}(bD)$}
\put(257.0,160.0){\footnotesize{\color[rgb]{0,.5,.5}$\delta$}}
\put(185.0,100.0){ $x_{2}$} \put(202.0,50.0){ \small$\delta$}
\put(217.0,43.0){ \small$\sqrt{\delta}$} \put(175.0,35.0){
\small$\sqrt{\delta}$} \put(290.0,30.0){$y_{1}$} \put(130.0,-5.0){$x_{1}$}
\put(195.0,-18.0){ $(c)$} \put(335.0,65.0){\small{Key}}
\put(360.0,47.0){\small{$S(w;\delta)$}}
\put(360.0,25.0){\small{$\widetilde{S}(w;\delta)$}}
\put(360.0,9.0){\small{$\\{\text{dist}(z,\mathcal{H}_{w}(bD))=\delta\\}$}}
\end{overpic} Figure 4. $3$D slices of the domain $D=\\{y_{2}>|z_{1}|^{2}\\}$
near $w=(0,0)$; $\mathcal{H}_{w}(bD)=\\{z_{2}=0\\}$. Here, $S(w;\delta)$ is
the “cap” $\\{z\in bD:\text{dist}(z,\mathcal{H}_{w}(bD))\leq\delta\\}$ and
$\widetilde{S}(w;\delta)$ is the projection of $S(w;\delta)$ onto
$\mathcal{T}_{w}(bD)=\\{y_{2}=0\\}$.
$(a)$ $D$ in the $x_{2}=0$ plane,
$(b)$ $D$ in the $x_{1}=0$ plane,
$(c)$ $\widetilde{S}(w;\delta)$ in the $y_{2}=0$ plane, i.e., in
$\mathcal{T}_{w}(bD)$.

Induced polyhedra are special cases of what are known as analytic polyhedra in
the complex analysis literature. However, as pointed out in [15, 16], the full
class of analytic polyhedra is not suitable for the kind of approximation
problems we are interested in. The size function may seem as an artefact of
the complex setting since it is absent in the construction of
$P_{\text{c}}(\varphi)$ above. However, convex polyhedra with nontrivial size
functions have appeared in the real setting; see Ludwig [26] and Glasauer and
Schneider [11, Section 5].

#### Random polyhedra and the metric of approximation.

We wish to discuss a random analogue of the following question investigated in
[16]: given a bounded strongly ${\mathbb{C}}$-convex domain
$D\subset\mathbb{C}^{d}$, determine the asymptotics as $n\rightarrow\infty$ of

(1.2) $v_{n}(D):=\inf\\{\lambda(D\setminus
P(\varphi;\boldsymbol{\delta})):\varphi=\\{w^{1},...,w^{n}\\}\subset bD,\
\boldsymbol{\delta}:bD\rightarrow(0,\infty),\
P(\varphi;\boldsymbol{\delta})\subset D\\},$

where $\lambda$ is the Lebesgue measure on $\mathbb{C}^{d}$. We know from [16]
that

(1.3) $v_{n}(D)\sim v^{\sharp}_{\mathbb{C}}(D)n^{-1/d}\qquad\text{as}\
n\to\infty,$

for some domain-dependent constant $v^{\sharp}_{\mathbb{C}}(D)$.

We randomize $P(\varphi;\boldsymbol{\delta})$ by taking $\varphi$ to be random
point sets. Let $\\{W^{j}\\}_{j\geq 1}$, be i.i.d. points sampled according to
a probability density $f(\cdot)$ with respect to the surface area measure
$\sigma$ on $bD$. We either take the random point set to be
$\varphi=\eta_{n}:=\\{W^{1},\ldots,W^{N_{n}}\\}$ (Poisson process) where
$N_{n}$ is an independent Poisson($n$) random variable, or
$\varphi=\beta_{n}:=\\{W^{1},\ldots,W^{n}\\}$ (binomial process). We shortly
comment on the choice of $\boldsymbol{\delta}$.

As with (1.2), we use the Lebesgue volume $\lambda$ to measure the error of
approximation of $D$ by $P(\varphi;\boldsymbol{\delta})$. Since we are
interested in randomly contained polyhedra but a random
$P(\varphi;\boldsymbol{\delta})$ (in the sense defined above) may not be
contained in $D$, we choose the following metric whereby the polyhedron is
penalised if it is not contained in $D$ (or is unbounded):

(1.4)
$\delta_{V}\big{(}D,P(\varphi;\boldsymbol{\delta})\big{)}:=\lambda(D\setminus
P(\varphi;\boldsymbol{\delta}))\mathbf{1}(P(\varphi;\boldsymbol{\delta})\subset
D)+\lambda(D)\mathbf{1}(P(\varphi;\boldsymbol{\delta})\subsetneq D).$

In the real case, such a penalty is unnecessary for inscribed polyhedra, but
is used for randomly circumscribed polyhedra; see Böröczky Jr. and Reitzner
[6]. An alternative would be the symmetric difference of $D$ and
$P(\varphi;\boldsymbol{\delta})$, as considered by Ludwig [26], Ludwig et al.
[27] and Grote and Werner [12] in the real case, which is a subject for future
investigation.

#### Overview of the results.

The first step towards understanding the random equivalent of $v_{n}(D)$ is to
investigate the asymptotics of
$\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))$ and
$\delta_{V}(D,P(\beta_{n};\boldsymbol{\delta}_{n}))$ for a given sequence of
functions $\boldsymbol{\delta}_{n}$. Since we want to study randomly contained
polyhedra, we assume that the polyhedra are contained in $D$ with high
probability, i.e., $\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset
D)\to 1$ as $n\to\infty$. For this, it is essential to assume that
$\boldsymbol{\delta}_{n}=g(z)(\frac{\log n}{n})^{1/d}$ for a function
$g:bD\to(0,\infty)$; see Proposition 2.6. For technical reasons, we assume
that $g$ is $\alpha$-Hölder for some $0<\alpha<1$.

Our first main result, Theorem 2.3, is that when $\varphi_{n}$ is either
$\eta_{n}$ (Poisson) or $\beta_{n}$ (binomial), and $\boldsymbol{\delta}_{n}$
is as above,

(1.5) $\Big{(}\frac{n}{\log
n}\Big{)}^{1/d}\delta_{V}(D,P(\varphi_{n};\boldsymbol{\delta}_{n}))\overset{p}{\rightarrow}\int_{bD}g(z)\text{d}\sigma(z),$

as $n\rightarrow\infty$, where $\overset{p}{\rightarrow}$ denotes convergence
in probability. Under further assumptions on the rate of convergence of the
containment probabilities
$\mathbb{P}(P(\varphi_{n};\boldsymbol{\delta}_{n})\subset D)$, we

  * $1.$

strengthen $\eqref{E:cvgprobintro}$ to $L^{1}$-convergence (Theorem 2.4),

  * $2.$

obtain variance (upper and lower) bounds for
$\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))$, and variance upper bounds
for $\delta_{V}(D,P(\beta_{n};\boldsymbol{\delta}_{n}))$ (Theorem 2.4), and

  * $3.$

bound the rate of normal approxiation of
$\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))$ (Theorem 2.5).

Our variance upper and lower bounds for
$\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))$ differ only by logarithmic
factors. We are unable to obtain good variance lower bounds for
$\delta_{V}(D,P(\beta_{n};\boldsymbol{\delta}_{n}))$. We prove a normal
approximation result for $\delta_{V}(D,P(\beta_{n};\boldsymbol{\delta}_{n}))$
assuming a suitable lower bound on the variance; see Theorem 2.5.

#### Optimal random approximations: a conjecture

In problems of approximation, it is of interest to compare the rate of random
approximations to that of optimal approximations. Note that if we ignore the
domain-dependent constant in (1.5), we obtain that the approximation rate is
$(n^{-1}\log n)^{1/d}$, which differs from the rate of optimal approximation
in (1.3) by a logarithmic factor. This additional factor is necessitated by
the need for the containment condition on the underlying random polyhedra. In
fact, once the containment condition is translated into a coverage condition
on the boundary (Lemma 3.14), the additional logarithmic factor is no longer
surprising in view of coverage results of Flatto and Newman [10], Hall [17]
and Janson [19] (although these do not apply directly to our setting).

Now, we compare the limiting constants in (1.3) and (1.5). At first glance, it
appears that the choice of density $f$ is absent in the limiting constant in
(1.5). But, it is implicit in the condition that
$\mathbb{P}(P(\varphi_{n};\boldsymbol{\delta}_{n})\subset D)\to 1$ as
$n\to\infty$. This raises the questions of: $(a)$ the best possible limiting
constant in (1.5), i.e. ,

$v_{D}^{*}(f)=\inf\left\\{\int_{bD}g(z)\text{d}\sigma(z):\mbox{$g$ is
continuous
and}\,\lim_{n\to\infty}\mathbb{P}(P(\varphi_{n};\boldsymbol{\delta}_{n})\subset
D)=1\right\\}$

for a fixed continuous density $f$, and $(b)$ the least possible constant,
$v_{{\mathbb{C}}}^{*}(D)$, over all such $f$. The proofs of [10, 17, 19] for
coverage results in the real setting suggest the following conjecture.

######  Conjecture 1.1.

Let $D\subset\mathbb{C}^{d}$ be a bounded strongly ${\mathbb{C}}$-convex
domain, and $f:bD\rightarrow(0,\infty)$ be a continuous probability density
function. Then,

(1.6)
$v^{*}_{D}(f)=(h_{d}\kappa_{2d-2})^{-1/d}\int_{bD}\frac{(16\nu_{D}(z))^{\frac{1}{2d}}}{f(z)^{\frac{1}{d}}}\,d\sigma(z),$

where $\nu_{D}$ is the complex-restricted curvature of $bD$ (see Definition
B.3), $\kappa_{d}$ is the Lebesgue volume of the unit ball in
$\mathbb{R}^{d}$, and
$h_{d}=2\int_{0}^{\pi/2}\cos^{d}\\!\theta\>\text{d}\theta$.

Note that the right-hand side of (1.6) is minimized when

$f=(16\nu_{D})^{\frac{1}{2d+2}}\left(\int_{bD}(16\nu_{D})^{\frac{1}{2d+2}}d\sigma\right)^{-1},$

i.e., $f\text{d}\sigma$ is the so-called normalized Möbius-Fefferman measure,
which appears in [16] in relation to $v_{{\mathbb{C}}}^{\sharp}(D)$, and thus
$v_{{\mathbb{C}}}^{*}(D)=(h_{d}\kappa_{2d-2})^{-1/d}\left(\int_{bD}(16\nu_{D})^{\frac{1}{2d+2}}d\sigma\right)^{\frac{d+1}{d}}.$
This is expected to differ from the constant $v^{\sharp}_{\mathbb{C}}(D)$ in
(1.3) only by a dimensional constant. See Section 2.3 for a remark on this
conjecture.

#### Comparison with real random polyhedra.

Needless to say, the literature on random real polyhedra is very large owing
to the numerous ways of constructing convex polyhedra. Nevertheless, there are
some recurring features, within the context of which we place our results.

As noted above, the exponent of $n$ in the rate of optimal approximations
(1.3) and random approximations (1.5) is the same, i.e., $2/2d$. This matching
of exponents in the optimal and random rates is seen in the (real) case of
approximations by $P_{\text{i}}(\varphi)$ and $P_{\text{c}}(\varphi)$ as well,
where the exponent is $2/(d-1)$. The quantities $2d$ and $d-1$ in the
exponents $1/d$ in the complex setting and $2/(d-1)$ in the real setting,
respectively, can be unified when interpreted in terms of the Hausdorff
dimensions of naturally-occuring metric space in these problems; see [15,
Section 1]. The appearance of the logarithmic factor in (1.5) is a point of
departure from the case of $P_{\text{i}}(\varphi)$ and
$P_{\text{c}}(\varphi)$. However, similar logarithmic factors appear even in
the real setting in the case of polyhedral constructions for which containment
needs to be imposed; e.g., see [11, Theorems 4 and 5]. Finally, though the
optimal and random limiting constants in the complex setting
($v^{\sharp}_{\mathbb{C}}(D)$ and $v^{*}_{D}$) are not as well-understood as
in many models of real polyhedra, we have some evidence to believe that they
too will only differ by a dimensional constant, as in the real case, e.g.,
compare the constants in [14] (best inscribed and circumscribed) and [39, 31,
6] (random inscribed and circumscribed).

So far, we have only placed the convergence in probability result (Theorem
2.3) into context. Our expectation, variance and normal approximation results
also compare well with what is known in the real setting, but we postpone the
details to Section 2.3.

#### Overview of the proofs.

The two key ideas underpinning our proofs are that the complement of an
induced polyehdron in $D$ can be expressed as a union of finitely many ‘cuts’
i.e., sets of the form $\left\\{z\in
D:\text{dist}(z,\mathcal{H}_{w}bD)<\boldsymbol{\delta}(w)\right\\}$, and that
a cut is well-approximated by a ‘tubular cut’. The volume of a union of
tubular cuts is then esitmated via a Weyl-type tube formula for non-uniform
tubes due to Roccaforte [36]. This requires analysis of an appropriate depth
function and the so-called visibility regions. The results on convergence in
probability and $L^{1}$-convergence follow from these estimates (and suitable
assumptions on the containment probabilities
$\mathbb{P}(P(\varphi_{n};\boldsymbol{\delta}_{n})\subset D)$). This approach
is in the spirit of [6], where Steiner’s tube formula is used to study volume
approximations of randomly circumscribed (real) polyhedra. For variance bounds
and normal approximation, we use the Efron–Stein, Poincaré, and second-order
Poincaré inequalities along with first-order estimates on the volumes of cuts
and tubular cuts. Some of these geometric estimates were already available in
the literature (Lemmas 3.2, 3.3, 3.5, 3.13 and 3.15), while the rest are
either new or are quantitatively precise versions of what was known
previously. We believe that these estimates are of independent interest in the
study of ${\mathbb{C}}$-convex domains.

##  2\. Limit theorems for random polyhedra

In this section, we formally introduce our model of random polyhedra and state
our main results. In Section 2.1, we define the class of polyhedral objects
under consideration. In Section 2.2, we state the main limit theorems for the
volume approximation. In Section 2.3, we make some observations related to the
main theorems. For reference, a list of notation (such as for complex
derivatives and gradients) is provided at the beginning of Section 3.

###  2.1. Polyhedra induced by strongly ${\mathbb{C}}$-convex domains

Recall that a set $K\subset\mathbb{R}^{d}$ is (real) convex if and only if the
intersection of $K$ with every real line is connected. In analogy with this, a
subset $D\subset\mathbb{C}^{d}$ is said to be ${\mathbb{C}}$-convex if for
every complex affine line $L$, $D\cap L$ is connected and simply connected,
i.e., the complement of $D$ in the extended line $L\cup\\{\infty\\}$ is
connected. A domain (i.e., a connected and open set) in ${\mathbb{C}}$ is
${\mathbb{C}}$-convex precisely when it is simply connected. In higher
dimensions, while all ${\mathbb{C}}$-convex domains in $\mathbb{C}^{d}$ are
topologically trivial, i.e., homeomorphic to the unit ball in
$\mathbb{C}^{d}$, the converse is far from true. ${\mathbb{C}}$-convex sets
satisfy many complex analogues of properties displayed by convex sets. As
mentioned in Section 1, ${\mathbb{C}}$-convexity is invariant under projective
(or linear fractional) transformations, i.e., mappings of the form (B.1). Akin
to the polar of a convex set, one may define the dual complement of a
${\mathbb{C}}$-convex set; see [1, Def. 2.1.1]. Moreover, any open or compact
${\mathbb{C}}$-convex set $D\subset\mathbb{C}^{d}$ satisfies a complex version
of the Hahn–Banach separation property: $\mathbb{C}^{d}\setminus D$ is a union
of complex hyperplanes in $\mathbb{C}^{d}$. We direct the reader to [1] for a
more detailed discussion of ${\mathbb{C}}$-convexity and related notions.

We now consider $\mathcal{C}^{2}$ domains, i.e., those of the form
$D=\\{z\in\mathbb{C}^{d}:\rho(z)<0\\}$, where $\rho$ is a $\mathcal{C}^{2}$
function on $\mathbb{C}^{d}$ whose gradient is nonvanishing on $bD$. Here,
$\rho$ is called a defining function of $D$. At any point $w\in bD$, the real
tangent space of $bD$ at $w$ is the $\mathbb{R}$-vector space

$\mathcal{T}_{w}bD=\\{v\in\mathbb{C}^{d}:\operatorname{Re}\left<\partial\rho(w),v\right>=0\\},$

while the complex tangent space of $bD$ at $w$ is the ${\mathbb{C}}$-vector
space

$\mathcal{H}_{w}bD=\\{v\in\mathbb{C}^{d}:\left<\partial\rho(w),v\right>=0\\}.$

We note that the complex tangent space at $w$ is the maximal subspace of the
real tangent space at $w$ that is invariant under the action of multiplication
by $i$.

Recall that $D\subset\mathbb{C}^{d}$ as above is real convex if and only if
the Hessian of $\rho$ when restricted to $\mathcal{T}_{w}bD$ is semi-positive
definite for every $w\in bD$. In the same way, $D\subset\mathbb{C}^{d}$ as
above is ${\mathbb{C}}$-convex if and only if the Hessian of $\rho$ when
restricted to $\mathcal{H}_{w}bD$ is semi-positive definite for every $w\in
bD$. This can be expressed in terms of complex derivatives as follows:

$Q_{w}(v,v):=\sum_{j,k=1}^{d}\dfrac{\partial^{2}{\rho}}{\partial{z_{j}}\partial{\overline{z}_{k}}}(w)v_{j}\overline{v}_{k}+\operatorname{Re}\sum_{j,k=1}^{d}\dfrac{\partial^{2}{\rho}}{\partial{z_{j}}\partial{z_{k}}}(w)v_{j}v_{k}\geq
0\quad\forall w\in bD,v\in\mathcal{H}_{w}bD.$

######  Definition 2.1.

Let $D\subset\mathbb{C}^{d}$, $d\geq 2$, and $\rho$ be as above. Then $D$ is
said to be strongly ${\mathbb{C}}$-convex if, for each $w\in bD$, $Q_{w}$ is
positive definite when restricted to $\mathcal{H}_{w}bD$.

Any such $D$ has the property that the complex tangent hyperplane
$w+\mathcal{H}_{w}bD=\\{z\in\mathbb{C}^{d}:\left<\partial\rho(w),w-z\right>=0\\}$
at any $w\in bD$ intersects $\overline{D}$ only at $w$; see [1, Sec. 2.5]. By
the continuity of the second-order derivatives, strong
${\mathbb{C}}$-convexity is equivalent to the existence of a continuous
function $c:bD\rightarrow(0,\infty)$ such that

(2.1) $Q_{w}(v,v)\geq c(w)||v||^{2}\quad\forall w\in
bD,v\in\mathcal{H}_{w}bD.$

Thus, when $D$ is bounded, strong ${\mathbb{C}}$-convexity is equivalent to
strong ${\mathbb{C}}$-linear convexity (in the sense of [22]), i.e., there
exists a $\tilde{c}>0$ such that

(2.2)
$\left|\left<\partial\rho(w),w-z\right>\right|\geq\tilde{c}||w-z||^{2}\quad\forall(z,w)\in\overline{D}\times
bD.$

See Proposition 3.1 (and the subsequent remarks) in [22]. In general, for a
fixed $w\in bD$, (2.1) implies (2.2) only for $z$ near $w$, but since the left
hand side of (2.2) is positive on $\overline{D}\setminus\\{w\\}$, the
compactness of $\overline{D}$ grants (2.2) for all $z\in\overline{D}$. In
geometric terms, $D$ is strongly ${\mathbb{C}}$-convex if $bD$ is locally
equivalent to a strongly real convex hypersurface in $\mathbb{C}^{d}$ via a
linear fractional transformation. For more insight into the geometry of such
domains, see Lemma 3.3, Lemma 3.13, and Appendix B.

Let $D\subset\mathbb{C}^{d}$ be a bounded strongly ${\mathbb{C}}$-convex
domain and $n\in{\mathbb{N}}$. Given a defining function $\rho$ of $D$,
$\varphi=\\{w^{1},...,w^{n}\\}\subset bD$ and a function
$\boldsymbol{\delta}:bD\rightarrow(0,\infty)$, let
$P(\varphi;\boldsymbol{\delta})$ be the union of those connected components of

(2.3)
$\bigcap_{j=1}^{n}\left\\{z\in\mathbb{C}^{d}:\frac{\left|\left<\partial\rho(w^{j}),w^{j}-z\right>\right|}{||\partial\rho(w^{j})||}>\boldsymbol{\delta}(w^{j})\right\\},$

that intersect $D$. To place (2.3) into context, we note that if
$K\subset\mathbb{R}^{d}$ is a $\mathcal{C}^{1}$ (real-)convex domain with
defining function $r$, then any $n$-faceted (real-)convex polyhedron
$P\subset\mathbb{R}^{d}$ can be written as

$P=\left\\{x\in\mathbb{R}^{d}:\left<\nabla
r(y^{j}),y^{j}-x\right>>\delta_{j},j=1,...,n\right\\}$

for some $y^{1},...,y^{n}\in bK$ and
$\delta_{1},...,\delta_{n}\in{\mathbb{R}}$. In particular, if
$\delta_{1}=\cdots=\delta_{n}=0$, then $P$ is the circumscribed polyhedron
$P_{\text{c}}(\varphi)$ for $\varphi=\\{y^{1},...,y^{n}\\}$.

######  Definition 2.2.

Given $D$ as above, an induced polyhedron is a nonempty set of the form
$P(\varphi;\boldsymbol{\delta})$. We say that $P(\varphi;\boldsymbol{\delta})$
is an induced polyhedron contained in $D$, if
$P(\varphi;\boldsymbol{\delta})\subset D$. We call $\varphi$ the source set
and $\boldsymbol{\delta}$ the size function of
$P(\varphi;\boldsymbol{\delta})$, respectively.

Note that, for any $w\in bD$, the set

$F_{w}:=\left\\{z\in\mathbb{C}^{d}:\frac{\left|\left<\partial\rho(w),w-z\right>\right|}{||\partial\rho(w)||}=\boldsymbol{\delta}(w)\right\\}$

is a real hypersurface in $\mathbb{C}^{d}$ that is foliated by complex
hyperplanes, and divides $\mathbb{C}^{d}$ into two unbounded
${\mathbb{C}}$-convex domains. Thus, $F_{w}$ can be viewed as a complex
analogue of a real hyperplane in $\mathbb{R}^{d}$. In view of this, we refer
to any nonempty set $F_{w_{j}}\cap bP(\varphi;\boldsymbol{\delta})$,
$j=1,...,n$, as a facet of $P(\varphi;\boldsymbol{\delta})$. The elements of
the collection

$\mathscr{P}_{n}(D):=\big{\\{}P(\varphi;\boldsymbol{\delta}):\varphi\subset
bD,\ |\varphi|\leq n,\ \text{and}\
\boldsymbol{\delta}:bD\rightarrow(0,\infty)\big{\\}}$

are called induced polyhedra with at most $n$ facets. This class is invariant
under complex affine transformations, i.e., given a source set $\varphi$ on
$bD$, size function $\boldsymbol{\delta}$, and
$A\in\operatorname{GL}(d;{\mathbb{C}})$, there exists a size function
$\boldsymbol{\delta}^{\prime}$ such that
$A(P(\varphi;\boldsymbol{\delta}))=P(A(\varphi);\boldsymbol{\delta}^{\prime})$.
This follows from (B.2).

###  2.2. Main results

We briefly introduce the Poisson and binomial processes, and then state our
main results for random polyhedra constructed on these point processes. See
Appendix A for a more formal definition of point processes and a list of
results regarding them that we use in this paper.

Let $f:bD\rightarrow(0,\infty)$ be a probability density function i.e.,
$\int_{bD}f(z)\text{d}\sigma(z)=1$. For $n\geq 1$, let $\eta_{n}$ be a Poisson
process with intensity measure $nf\sigma$. From the standard representation of
a Poisson process, we can assume that $\eta_{n}=\\{W^{i}\\}_{1\leq i\leq
N_{n}}$, where $W^{i}$ are i.i.d. with probability density $f$, and $N_{n}$ is
an independent Poisson random variable with mean $n$. A second point process
of interest is the binomial process. We say that a process $\beta_{n}$ is a
binomial process on $bD$ with $n$ points and density $f$ if
$\beta_{n}=\\{W^{1},\ldots,W^{n}\\}$ where, once again, $W^{i}$ are i.i.d.
points with probability distribution $f\sigma$.

For a random polyhedral approximation of $D$, we consider an induced
polyhedron whose source set is a point process on $bD$ (either $\eta_{n}$ or
$\beta_{n}$) and size function is
$\boldsymbol{\delta}=\boldsymbol{\delta}_{n}$, where we impose the following
assumption on $\boldsymbol{\delta}_{n}$:

(2.4)
$\boldsymbol{\delta}_{n}(w)=g(w)\delta_{n}{,}\quad\mbox{where}\quad\delta_{n}:=\left(\frac{\log
n}{n}\right)^{\frac{1}{d}}$

and $g$ is a fixed positive function on $bD$ such that
$g\in\mathcal{C}^{\alpha}(bD)$, for some $\alpha\in(0,1]$.

Our main objective in this paper is to understand the asymptotics of the
volume of the portion of $D$ that is left uncovered by the random polyhedra
$P(\eta_{n};\boldsymbol{\delta}_{n})$ or
$P(\beta_{n};\boldsymbol{\delta}_{n})$ as $n\rightarrow\infty$. Two of our
main results are convergence in probability, and under further assumptions,
$L^{1}$-convergence of the random variables
$\delta_{V}(D,P(\eta_{n},\boldsymbol{\delta}_{n}))$ and
$\delta_{V}(D,P(\beta_{n},\boldsymbol{\delta}_{n}))$, where recall from (1.4)
that

$\delta_{V}(D,P)=\lambda(D\setminus P)\mathbf{1}(P\subset
D)+\lambda(D)\mathbf{1}(P\subsetneq D).$

The rest are variance bounds and normal approximation for these random
variables.

######  Theorem 2.3 (Convergence in Probability).

Let $f:bD\rightarrow(0,\infty)$ be a bounded probability density function that
is bounded away from $0$, and $\boldsymbol{\delta}_{n}=g\delta_{n}$ be as in
(2.4). Further, assume that $g$ is such that

(2.5) $\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset D)\rightarrow
1\mbox{ as $n\rightarrow\infty$}.$

Then,

(2.6) ${\left(\frac{n}{\log
n}\right)^{1/d}}\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\int_{bD}g(z)\text{d}\sigma(z)\mbox{ as
$n\rightarrow\infty$}.$

If (2.5) holds for $\beta_{n}$, then (2.6) holds for $\beta_{n}$.

The above theorem gives a simple condition under which the error in volume
approximation decays to zero at a fixed rate as the number of facets of the
random polyhedra increase. While (2.6) gives the precise rate of volume
approximation for a fixed $f$ and $g$, it leaves open the question of the the
best possible rate of volume approximation among all random polyhedra; see
Conjecture 1.1.

######  Theorem 2.4 ($L^{1}$-Convergence and Variance Bounds).

Let $f:bD\rightarrow(0,\infty)$ be a bounded probability density function that
is bounded away from $0$, and $\boldsymbol{\delta}_{n}=g\delta_{n}$ be as in
(2.4). Further, assume that $g$ is such that

(2.7) $\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset
D)=1-o\left(n^{-\left(1+\frac{2}{d}+\epsilon\right)}\right)\ \text{as}\
n\rightarrow\infty,$

for some $\epsilon>0$. Then,

(2.8) $\lim_{n\to\infty}\mathbb{E}\left|{\left(\frac{n}{\log
n}\right)^{1/d}}\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))-\int_{bD}g(z)\text{d}\sigma(z)\right|=0$

and there exist $c_{1},c_{2}>0$ such that

(2.9) $c_{1}{\frac{1}{n^{1+2/d}(\log
n)^{2+2/d}}}\leq\operatorname{{\mathbb{V}ar}}[\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))]\leq
c_{2}{\frac{(\log n)^{2+2/d}}{n^{1+2/d}}},\quad n\geq 1.$

If (2.7) holds for $\beta_{n}$, then (2.8) and the upper bound in (2.9) hold
for $\beta_{n}$.

Note that the assumption in (2.7) is to ensure that the dominating
contribution is from the volume of the random polyhedron and not the non-
containment event. Our proofs give explicit bounds in terms of non-containment
probabilities, and it may be possible to weaken (2.7). For instance, a quick
inspection of the proof of Theorem 2.4 shows that to obtain the expectation
asymptotics in (2.8), it suffices to assume that the non-containment
probability is $o\Big{(}(\frac{\log n}{n})^{1/d}\Big{)}$ as
$n\rightarrow\infty$, but we have assumed (2.7) to give a combined result for
expectation and variance.

To state a normal approximation result, we fix a metric of approximation for
random variables. Given two random variables, $X$ and $Y$, the Wasserstein
distance $d_{W}(X,Y)$ is defined as

(2.10)
$d_{W}(X,Y):=\sup_{h\in\mathrm{Lip}(1)}|\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]|,$

where $\mathrm{Lip}(1)$ is the class of all functions
$h:{\mathbb{R}}\to{\mathbb{R}}$ with Lipschitz constant at most $1$. The
Wasserstein distance metrizes the topology of weak convergence i.e., if
$d_{W}(X_{n},X)\to 0$ as $n\to\infty$ then $X_{n}\stackrel{{\scriptstyle
D}}{{\longrightarrow}}X$ as $n\to\infty$ where $\stackrel{{\scriptstyle
D}}{{\longrightarrow}}$ denotes convergence in distribution or weak
convergence. Henceforth, $Z$ denotes a standard normal variable.

######  Theorem 2.5 (Central Limit Theorem).

Let $f:bD\rightarrow(0,\infty)$ be a bounded probability density function that
is bounded away from $0$, and $\boldsymbol{\delta}_{n}=g\delta_{n}$ be as in
(2.4). Further, assume that $g$ is such that (2.7) holds for some
$\epsilon>0$. Then,

(2.11)
$d_{W}\left(\frac{\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))-{\mathbb{E}}[\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))]}},Z\right)\leq
C{\frac{(\log n)^{6+6/d}}{n^{\frac{\min\\{\epsilon,1\\}}{2}}}},\quad n\geq 1.$

If (2.7) and the lower bound in (2.9) hold for $\beta_{n}$, then (2.11) holds
for $\beta_{n}$.

We have made an assumption on the decay of $\delta_{n}$ in (2.4), and on the
asymptotic behavior of containment probabilities, as in (2.5) and (2.7). The
following lemma assures us that these assumptions are compatible as well as
essential.

######  Proposition 2.6.

Let $f:bD\rightarrow(0,\infty)$ be a bounded probability density function that
is bounded away from $0$, and $\boldsymbol{\delta}_{n}=g\delta_{n}$ with
$\delta_{n}$ as in (2.4) and $g$ is a continuous function. Then, the following
hold.

  * (a)

There exists a $c_{D}>0$ such that if $\min_{z\in bD}g(z)<c_{D}$, then

$\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset D)\rightarrow 0\mbox{
as $n\rightarrow\infty$}.$

  * (b)

For any $\alpha>0$, there exists a $C_{D}>0$ such that if $\min_{z\in
bD}g(z)>C_{D}$, then

(2.12) $\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset
D)=1-{\mathcal{O}}(n^{-\alpha})\mbox{ as $n\rightarrow\infty$}.$

  * (c)

Both (a) and (b) also hold when $\eta_{n}$ is replaced by $\beta_{n}$.

It is clear from our proof that Part $(b)$ holds more generally for any
measurable $g$. A more careful analysis of our proofs can yield estimates on
$c_{D},C_{D}$ but given that these are likely to be far from the sharp bounds,
we avoid quantifying them here.

###  2.3. Remarks

We now remark in detail about the assumptions in our results, the nature of
results and how they compare to real inscribed polyhedra.

Containment condition. Theorem 2.3 assumes that containment happens with high
probability i.e., (2.5) holds. Suppose not i.e.,
$\liminf_{n\to\infty}\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset
D)<1$. Then we can show that along some subsequence,
$\delta_{V}(D,P(\eta_{n};\boldsymbol{\delta}_{n}))$ converges in probability
to a non-degenerate random variable supported on $\\{0,\lambda(D)\\}$. Thus,
the error doesn’t converge to $0$ if (2.5) is violated.

Optimizing the size function. As mentioned earlier, non-trivial size functions
occur in similar models in the real case. However, in the real case, one is
able to obtain the decay rate of the optimal $\boldsymbol{\delta}_{n}$ with
explicit constants because the transition from containment probability to
coverage probability on the boundary is done via a suitable Riemannian metric,
following which the probabilities are estimated using the classical coverage
results of [10, 17, 19]. In our case, the rate of decay of
$\boldsymbol{\delta}_{n}$, i.e., Lemma 2.6, is also obtained by reducing the
question of containment to that of coverage of the boundary by the so-called
caps of the approximating polyhedra. However, due to the inhomogeneous nature
of the caps, this is at best a coverage problem in a sub-Riemannian metric,
for which no coverage results of the type in [10, 17, 19] are available. This
is one of the obstacles in proving Conjecture 1.1.

Comparison with real polyhedra. In Section 1.1, we have already compared
Theorem 2.3 with the results known for randomly inscribed and circumscribed
polyhedra in the real setting. We now compare the other limit theorems in this
paper with those for randomly inscribed polyhedra (see [31, 32, 34, 35, 41,
43]). Recall that in these models, $K\subset\mathbb{R}^{d}$ is a smooth convex
body, the polyhedron, say $P_{n}$, is constructed as the convex hull of a
point process (Poisson or binomial) on $bK$, and one studies the asymptotics
of $V_{n}:=\lambda(K\setminus P_{n})$. It is known that

  * $(a)$

$\mathbb{E}[V_{n}]\sim v^{*}_{{\mathbb{R}}}n^{-2/(d-1)}$ as
$n\rightarrow\infty$ ([31, Theorem 1]),

  * $(b)$

$\operatorname{{\mathbb{V}ar}}[V_{n}]=\Theta(n^{-1-4/(d-1)})$ as
$n\rightarrow\infty$ ([32, Theorem 8] and [35, Theorem 1.1]),

  * $(c)$

$d_{K}\left(\frac{V_{n}-\mathbb{E}[V_{n}]}{\sqrt{BV[V_{n}]}},Z\right)\leq
Cn^{-1/2}(\log n)^{3(1+2/(d-1))}$ ([41, Theorem 1 and Section 4.1]),

where $d_{K}$ is the Kolmogorov distance and $v^{*}_{{\mathbb{R}}}$ is an
explicit constant described in terms of the Blaschke surface area measure. Up
to logarithmic factors, our expectation asymptotics and variance bounds
resemble the above results with the exponent $(d-1)$ replaced by $2d$. The
error in normal approximation has an extra $\epsilon$ dependence owing to
assumption (2.7). An expectation result similar to $(a)$ is proven for
randomly circumscribed polytopes in [6], but we are not aware of variance
bounds or a central limit theorem for this model. Similar results are known if
the point process (Poisson or binomial) is in $K$ (see [33]), where the
exponent is $(d+1)$ instead of $(d-1)$. Exact variance asymptotics have also
been obtained for real random polytopes in [8, Corollary 1.1 and 1.2] (see
also [40, Chapter 5]) via scaling transforms.

On the metric of normal approximation. Our proof for the normal approximation
result (Theorem 2.5) uses the second-order Poincaré inequalities for
Wasserstein distance in [24, 21] (see Appendix A). One can instead use the
second-order Poincaré inequalities for the stronger Kolmogorov distance from
[24, 21] but this involves bounding some additional terms. Though this should
be possible using our current estimates, we avoid doing so here. We expect
that the rate of convergence will be of same order even under Kolmogorov
distance.

##  3\. Geometric estimates

In this section, we lay the groundwork for the deterministic aspects of our
main proofs. Subsection 3.1 contains some new definitions that will be used
throughout the paper. The subsequent subsections contain some key local
estimates. For a quick summary of these estimates, the reader may jump to the
statements of Theorems 3.7 and 3.10 and Propositions 3.16 and 3.17. We also
note that the assumption of $\alpha$-Hölder continuity on $g$ is only
essential to the proofs of (3.6) in Theorem 3.7, Corollary 3.12, and
Proposition 3.17.

######  Notation.

The following notation will be used throughout the paper.

  1. (i)

For a point $z\in\mathbb{C}^{d}$

     * •

$(z_{1},...,z_{d})$ denotes $z$ in complex coordinates,

     * •

$(x_{1},y_{1},...,x_{d},y_{d})$ denotes $z$ in real coordinates, i.e.,
$z_{j}=x_{j}+iy_{j}$, $j=1,...,d$,

     * •

$z^{\prime}=(z_{1},...,z_{d-1})\in{\mathbb{C}}^{d-1}$ denotes the projection
of $z$ onto the space $\\{z:z_{d}=0\\}$,

     * •

$||z||$ denotes the Euclidean norm of $z$.

  2. (ii)

For $z,w\in\mathbb{C}^{d}$, $\left<z,w\right>=z_{1}w_{1}+\cdots+z_{d}w_{d}$.
Thus, $||z||=\left<z,\overline{z}\right>^{1/2}$.

  3. (iii)

For $z\in\mathbb{R}^{d}$ and $r>0$, $B_{w}(r)$ denotes the Euclidean ball of
radius $r$ centered at $z$.

  4. (iv)

$\lambda$ and $\lambda_{2d-1}$ denote the Lebesgue volume measures on
$\mathbb{C}^{d}\cong\mathbb{R}^{2d}$ and
${\mathbb{C}}^{d-1}\times\mathbb{R}\cong\mathbb{R}^{2d-1}$, respectively.

  5. (v)

For a $\mathcal{C}^{1}$ domain $D\subset\mathbb{C}^{d}$,

     * •

$\hat{\eta}(w)=(\hat{\eta}_{1}(w),...,\hat{\eta}_{d}(w))$ denotes the inner
unit normal to $bD$ at $w\in bD$,

     * •

$\sigma$ denotes the Euclidean surface area measure on $bD$.

  6. (vi)

$\operatorname{GL}(m;{\mathbb{C}})$ denotes the space of $m\times m$
invertible complex matrices.

  7. (vii)

For a $\mathcal{C}^{1}$ function $f:U\rightarrow\mathbb{R}$ on an open set
$U$,

     * •

$\partial
f=\left(\frac{\partial{f}}{\partial{z_{1}}},\cdots,\frac{\partial{f}}{\partial{z_{d}}}\right)=\left(\frac{1}{2}\frac{\partial{f}}{\partial{x_{1}}}-\frac{i}{2}\frac{\partial{f}}{\partial{y_{1}}},\cdots,\frac{1}{2}\frac{\partial{f}}{\partial{x_{d}}}-\frac{i}{2}\frac{\partial{f}}{\partial{y_{d}}}\right)$,
when $U\subset\mathbb{C}^{d}$,

     * •

$\nabla
f=\left(\frac{\partial{f}}{\partial{x_{1}}},\frac{\partial{f}}{\partial{y_{1}}},...,\frac{\partial{f}}{\partial{x_{d}}},\frac{\partial{f}}{\partial{y_{d}}}\right)$,
when $U\subset\mathbb{C}^{d}$,

     * •

$\nabla
f=\left(\frac{\partial{f}}{\partial{x_{1}}},\frac{\partial{f}}{\partial{y_{1}}},...,\frac{\partial{f}}{\partial{x_{d-1}}},\frac{\partial{f}}{\partial{y_{d-1}}},\frac{\partial{f}}{\partial{x_{d}}}\right)$,
when $U\subset{\mathbb{C}}^{d-1}\times\mathbb{R}$.

  8. (viii)

Given a compact set $X\subset\mathbb{C}^{d}$ and $\alpha\in(0,1)$, the space
of $\alpha$-Hölder functions on $X$ is

$\mathcal{C}^{\alpha}(X)=\left\\{g:X\rightarrow\mathbb{R}:||g||_{\alpha}:=||g||_{\infty}+\sup\limits_{\begin{subarray}{c}x,y\in
X\\\ x\neq
y\end{subarray}}\frac{|g(x)-g(y)|}{||x-y||^{\alpha}}<\infty\right\\},$

where $||g||_{\infty}=\sup_{x\in X}|g(x)|$.

  9. (ix)

Given a family of functions $\\{f_{w}\\}_{w\in bD}$ and a non-negative
function $h$ on a normed space $\mathbb{X}$, $f_{w}=O_{D}(h)$ as
$||x||_{\mathbb{X}}\rightarrow 0$ denotes the existence of
$M_{D},\delta_{D}>0$ such that

$|f_{w}(x)|\leq M_{D}h(x)\quad\text{for all}\ w\in
bD,||x||_{\mathbb{X}}<\delta_{D}.$

The notation $f_{w}^{1},f_{w}^{2}=O_{D}(h)$ as $||x||\rightarrow 0$ simply
denotes that $f_{w}^{1}=O_{D}(h)$ as $||x||\rightarrow 0$ and
$f_{w}^{2}=O_{D}(h)$ as $||x||\rightarrow 0$.

  10. (x)

Given real-valued functions $f,h$ on some set $S$, we say that $f\lesssim h$
if $f(x)\leq Ch(x)$ for some $C$ independent of $x\in S$. We abbreviate the
statement $f\lesssim h\lesssim f$ to $f\approx h$.

###  3.1. Preliminaries

In this subsection, we introduce some sets and measurements associated to
induced polyhedra. These will make crucial appearances in our main proofs. See
Figure 5 for an illustrative picture of some of these sets and measurements.
Towards the end, we also collect some useful results from the literature.

######  Definition 3.1.

Let $D\subset\mathbb{C}^{d}$, $d\geq 2$, be a bounded strongly
${\mathbb{C}}$-convex $\mathcal{C}^{2}$ domain, and
$\rho:\mathbb{C}^{d}\rightarrow\mathbb{R}$ be a $\mathcal{C}^{2}$ defining
function of $D$. Let $\boldsymbol{\delta}:bD\rightarrow(0,\infty)$ be a
continuous function and $\delta>0$.

  1. $(a)$

For any set $S\subseteq bD$ and bounded function $g:S\rightarrow[0,\infty)$,
the $g$-tube of $S$ is the set

$T_{g}(S):=\\{w+t\hat{\eta}(w):w\in S,\ 0\leq t\leq g(w)\\},$

where the notation is abbreviated to $T_{c}(S)$ when $g|_{S}\equiv c$.

  2. $(b)$

The function $L:\mathbb{C}^{d}\times bD\rightarrow{\mathbb{C}}$ is defined as

$L(z,w)=L_{\rho}(z,w):=\left<\frac{\partial\rho(w)}{||\partial\rho(w)||},w-z\right>=\frac{\left<\partial\rho(w),w-z\right>}{||\partial\rho(w)||}.$

Note that $\\{z\in\mathbb{C}^{d}:L(z,w)=0\\}$ is $\mathcal{H}_{w}bD$, the
complex tangent plane to $bD$ at $w$.

  3. $(c)$

The $\delta$-cut at $w\in bD$ is the (open) set

$C(w;\delta):=\left\\{z\in D:|L(z,w)|<\delta\right\\}.$

  4. $(d)$

The $\delta$-cap at $w\in bD$ is the (closed) set

$S(w;\delta):=\overline{C(w;\delta)}\cap bD=\left\\{\zeta\in
bD:|L(\zeta,w)|\leq\delta\right\\}.$

  5. $(e)$

For $w,\zeta\in bD$, the inner depth of $C(w;\delta)$ at $\zeta$ is

$r_{w}(\zeta;\delta):=\sup\\{s\geq
0:\zeta+t\hat{\eta}(\zeta)\in\overline{C(w;\delta)},\ \text{for all}\ 0\leq
t\leq s\\},$

and the maximum ${\boldsymbol{\delta}}$-visibility at $\zeta$ is

$r_{\infty}(\zeta;\boldsymbol{\delta}):=\sup\\{r_{w}(\zeta;\boldsymbol{\delta}(w)):w\in
bD\\}.$

  6. $(f)$

The outer depth of $C(w;\delta)$ is

$R_{w}(\delta):=\inf\\{r\geq 0:\overline{C(w;\delta)}\subset T_{r}(bD)\\}.$

and $R_{\infty}(\boldsymbol{\delta})=\sup\\{R_{w}(\boldsymbol{\delta}(w)):w\in
bD\\}$.

  7. $(g)$

The tubular $\delta$-cut at $w\in bD$ is the set

$TC(w;\delta):=T_{r_{w}(\cdot;\delta)}(S(w;\delta))=\left\\{\zeta+t\hat{\eta}(\zeta):\zeta\in
S(w;\delta),\ 0\leq t\leq r_{w}(z;\delta)\right\\}.$

  8. $(h)$

For $s\geq 0$, the $(\boldsymbol{\delta},s)$-visibility region at $\zeta\in
bD$ is the set

$G_{s}(\zeta;\boldsymbol{\delta}):=\\{w\in
bD:r_{w}(\zeta;\boldsymbol{\delta}(w))\geq s\\}=\left\\{w\in
bD:|L(\zeta+t\hat{\eta}(\zeta),w)|\leq\boldsymbol{\delta}(w)\ \text{for}\
t\in[0,s]\right\\}.$

\begin{overpic}[grid=false,tics=10,scale={0.5}]{drawing-3.pdf}
\put(187.0,51.0){\scriptsize Key} \put(200.0,39.0){\scriptsize$C(w;\delta)$}
\put(200.0,29.0){\scriptsize$TC(w;\delta)$}
\put(200.0,19.0){\scriptsize$bC(w;\delta)\cap D$}
\put(200.0,10.0){\scriptsize$S(w;\delta)$}
\put(200.0,0.0){\scriptsize$z+r_{w}(z;\delta)\hat{\eta}(z)$}
\put(9.0,68.0){\rotatebox{56.0}{\scriptsize$R_{w}(\delta)$}}
\put(50.0,27.0){$z$} \put(83.0,20.0){$w$} \put(335.0,20.0){$w$} \end{overpic}
Figure 5. A schematic of a $\delta$-cut, $\delta$-cap and tubular $\delta$-cut
with source $w$. In the figure on the right, the curve in teal and the region
in pink are the portions of $bC(w;\delta)$ and $C(w;\delta)$, respectively,
that are not contained in $TC(w;\delta)$.

The above definitions are independent of the choice of defining function for
$D$ (see Lemma 3.2 below). For a fixed
$\boldsymbol{\delta}:bD\rightarrow(0,\infty)$, and $w\in bD$,
$C(w;\boldsymbol{\delta})$, $S(w;\boldsymbol{\delta})$,
$TC(w;\boldsymbol{\delta})$, $r_{w}(\cdot;\boldsymbol{\delta})$ and
$R_{w}(\boldsymbol{\delta})$ denote $C(w;\boldsymbol{\delta}(w))$,
$S(w;\boldsymbol{\delta}(w))$, $TC(w;\boldsymbol{\delta}(w))$,
$r_{w}(\cdot;\boldsymbol{\delta}(w))$ and $R_{w}(\boldsymbol{\delta}(w))$,
respectively. Lastly, note that for all $w,\zeta\in bD$,

  * $*$

$r_{w}(\zeta;\boldsymbol{\delta}(w))=-\infty$ if $\zeta\notin
S(w;\boldsymbol{\delta}(w))$;

  * $*$

$TC(w;\boldsymbol{\delta}(w))\subseteq\overline{C(w;\boldsymbol{\delta}(w))}\subseteq
T_{R_{\infty}(\boldsymbol{\delta})}(bD)$;

  * $*$

$G_{0}(\zeta;\boldsymbol{\delta})=\\{w\in bD:\zeta\in
S(w;\boldsymbol{\delta}(w))\\}$, and
$G_{s}(\zeta;\boldsymbol{\delta})=\emptyset$ if
$s>r_{\infty}(\zeta;\boldsymbol{\delta})$.

The following two lemmas allow us to (locally) simplify the defining function
of $D$. These results are well-known, but fragmented throughout the
literature. For the reader’s convenience, we state them here and relegate
their proofs to Appendix B.

######  Lemma 3.2.

Let $D$ and $\rho$ be as in Definition 3.1.

  * $(a)$

For any defining function $\widetilde{\rho}$ of $D$,
$L_{\widetilde{\rho}}(z,w)=L_{\rho}(z,w)$ for all
$(z,w)\in\mathbb{C}^{d}\times bD$.

  * $(b)$

For $M\in\operatorname{GL}(d;{\mathbb{C}})$, if $\rho^{*}$ denotes the
defining function $\rho\circ M^{-1}$ of $M(D)$, then

$L_{\rho}(z,w)=\frac{||\partial\rho^{*}(M(w))||}{||\partial\rho(w)||}L_{\rho^{*}}(M(z),M(w)),\quad
z\in\mathbb{C}^{d},w\in bD.$

In particular, if $M$ is unitary, then
$||\partial\rho^{*}(M(w))||=||\partial\rho(w)||$ for all $w\in bD$. Thus, the
cuts, caps, depths, and visibility regions are preserved by $M$.

######  Lemma 3.3.

Let $D$ be as in Definition 3.1. Then there exists an $\varepsilon_{D}>0$ so
that for each $w\in bD$, there is a unitary transformation $U_{w}$ on
$\mathbb{C}^{d}$ such that $U_{w}(w)=0$, and
$\alpha_{j},\beta_{j},e\in\mathbb{R}$ and $\gamma_{j}\in{\mathbb{C}}$ such
that

  * $(a)$

$\alpha_{j},\beta_{j},e,$ and $\gamma_{j}$ depend continuously on $w$;

  * $(b)$

$0\leq\beta_{j}<\alpha_{j}$ for $j=1,...,d-1$; and

  * $(c)$

$D^{w}:=B_{0}(\varepsilon_{D})\cap
U_{w}(D)=\left\\{(z^{\prime},x_{d}+iy_{d})\in
B_{0}(\varepsilon_{D}):y_{d}>f_{w}(z^{\prime},x_{d})\right\\}$ for some
$\mathbb{R}$-valued $f_{w}$ such that

$f_{w}(z^{\prime},x_{d})=\sum_{j=1}^{d-1}\alpha_{j}|z_{j}|^{2}+\sum_{j=1}^{d-1}\beta_{j}\operatorname{Re}(z_{j}^{2})+\operatorname{Im}\sum_{j=1}^{d-1}\gamma_{j}z_{j}x_{d}+ex_{d}^{2}+O_{D}(||(z^{\prime},x_{d})||^{3})$

as $||(z^{\prime},x_{d})||\rightarrow 0$.

In particular, the complex-restricted curvature of $D$ at $w$ is
$\nu_{D}(w)=\frac{1}{16}\prod_{j=1}^{d-1}(\alpha_{j}^{2}-\beta_{j}^{2})$.

######  Remark 3.4.

In the subsequent subsections, the above lemma will be often used to reduce
any local computation at a $w\in bD$ to a corresponding computation for
$D^{w}$ at $0$. In this case, for all $z\in
bD^{w}=\left\\{(z^{\prime},x_{d}+iy_{d})\in
B_{0}(\varepsilon_{D}):y_{d}=f_{w}(z^{\prime},x_{d})\right\\}$,

$L(0,z)=\left<(0,...,0,i),z\right>=iz_{d}\qquad\text{and}\qquad\hat{\eta}(z)=\frac{(-\nabla
f_{w}(z^{\prime},x_{d}),1)}{\sqrt{1+||\nabla f_{w}(z^{\prime},x_{d})||^{2}}}.$

The above quantities are independent of the choice of defining function for
$D^{w}$ (see Lemma 3.2) and, hence, can be computed by taking a particular
choice, e.g., $\rho_{w}(z):=f_{w}(z^{\prime},x_{d})-y_{d}$. Also, note that
for $z=(z^{\prime},z_{d})\in bD^{w}$,

(3.1)
$y_{d}-x_{d}\dfrac{\partial{f_{w}}}{\partial{x_{d}}}(z^{\prime},x_{d})=f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})-ex_{d}^{2}+O_{D}(||(z^{\prime},x_{d})||^{3})\qquad\text{as}\
||(z^{\prime},x_{d})||\rightarrow 0,$

where
$f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})=\sum_{j=1}^{d-1}\alpha_{j}|z_{j}|^{2}+\sum_{j=1}^{d-1}\beta_{j}\operatorname{Re}(z_{j}^{2})$.
Due to the strong ${\mathbb{C}}$-convexity of $D$ — see (2.1), there is a
$c_{D}>0$ (independent of $w\in bD$) such that

(3.2)
$Q_{0}^{D^{w}}(z^{\prime},z^{\prime})=f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})\geq
c_{D}||z^{\prime}||^{2},\quad\text{for all}\
z^{\prime}\in{\mathbb{C}}^{d-1}\cong\mathcal{H}_{0}bD^{w}.$

In our setting, $|L(z,w)|$ is the distance of $z\in\overline{D}$ from the
affine space $w+\mathcal{H}_{w}bD$, $w\in bD$. The following estimates on
$|L(z,w)|$ from [22] are useful.

######  Lemma 3.5.

Let $D$ be as in Definition 3.1. Then,

(3.3) $||w-z||^{2}\lesssim|L(z,w)|\lesssim||w-z||,\quad z\in\overline{D},w\in
bD.$

Moreover, there exists a $t_{D}>0$ such that,

(3.4) $|L(\zeta,w)|+t\approx|L(\zeta+t\hat{\eta}(\zeta),w)|,\quad\zeta,w\in
bD,t<t_{D}.$

###### Proof.

The first estimate (3.3) is essentially proved at the end of [22, Section
3.3]. Since it is only stated for $z,w\in bD$ therein, we recall the argument
here. The upper bound follows easily since
$|L(z,w)|=\left\|\left<\frac{\partial\rho(w)}{||\partial\rho(w)||},w-z\right>\right\|\leq||w-z||$.
The lower bound is the strong ${\mathbb{C}}$-convexity condition; see (2.2).

The second estimate (3.4) is exactly [22, Lemma 4.3]. ∎

Lastly, we recall Roccaforte’s variation of Weyl’s tube formula for nonuniform
tubes. The theorem is originally formulated for smooth oriented hypersurfaces
in $\mathbb{R}^{d}$, but we state it for boundaries of $\mathcal{C}^{2}$
domains in $\mathbb{R}^{2d}$, as these are the relevant hypersurfaces in our
case. We note that Roccaforte’s proof does not require higher than
$\mathcal{C}^{2}$-regularity.

######  Theorem 3.6 ([36, Theorem 1]).

Let $D\subset\mathbb{R}^{2d}$ be a bounded $\mathcal{C}^{2}$ domain. Let
$g:bD\rightarrow[0,\infty)$ be a bounded, measurable function. Then, for
$\sup_{bD}|g|$ sufficiently small,

$\lambda(T_{g}(bD))=\sum_{j=0}^{d-1}\frac{(-1)^{j}}{j+1}\int_{bD}g(x)^{j+1}s_{j}(x)\,d\sigma(x),$

where $s_{j}(x)$ is the $j^{\text{th}}$ symmetric polynomial in the principal
curvatures (eigenvalues of the shape operator) of $bD$ at $x$. Here, $bD$ is
oriented according to the inner unit normal field.

###  3.2. Depth estimates

In view of Theorem 3.6 above, it is useful to estimate the inner and outer
depths of the cuts of an induced polyhedron. This allows us to squeeze the
complement of the polyhedron in $D$ between two (inner) tubular neighborhoods
of the boundary of $D$, one uniform and one nonuniform.

######  Theorem 3.7.

Let $D$ be as in Definition 3.1, $\delta>0$, and
$\boldsymbol{\delta}=g\delta$, for some positive function
$g\in\mathcal{C}^{\alpha}(bD)$, where $\alpha\in(0,1)$. There exist
$C_{D},\delta_{D}>0$ such that, if $\zeta\in bD$ and
$||\boldsymbol{\delta}||_{\infty}<\delta_{D}$, then

(3.5) $\displaystyle R_{\infty}(\boldsymbol{\delta})\leq
C_{D}||\boldsymbol{\delta}||_{\infty},$ (3.6) $\displaystyle 0\leq
r_{\infty}(\zeta;\boldsymbol{\delta})-g(\zeta)\delta\leq
C_{D}||g||_{\alpha}^{2}\delta^{1+\frac{\alpha}{2}}.$

To prove the above theorem, we need estimates on $r_{w}(\zeta;\delta)$. We
already have that $r_{w}(\zeta;\delta)\lesssim\delta-|L(\zeta,w)|$ for
$\zeta\in S(w;\delta)$ from (3.4). This does not suffice for our purposes, and
we improve this estimate as follows.

######  Lemma 3.8.

Let $D$ be as in Definition 3.1. There exist $C_{D},\delta_{D}>0$ such that
for any $w\in bD$, $\delta<\delta_{D}$, and $z\in S(w;\delta)$,

(3.7) $\delta-|L(z,w)|\leq
r_{w}(z;\delta)\leq\sqrt{\delta^{2}-|L(z,w)|^{2}}+C_{D}\delta^{2}.$

###### Proof.

By Lemmas 3.2 and 3.3, we may assume that $w=0$ and $D=D^{w}$. We denote
$f_{w}$ by $f$, $\tfrac{\partial f}{\partial x_{d}}(z^{\prime},x_{d})$ by
$\partial_{x_{d}}f(z)$, and the $d$-th component of $\hat{\eta}(z)$ by
$\hat{\eta}_{d}(z)$. We will invoke the properties of $D^{w}$ and $f_{w}$
collected in Remark 3.4.

Owing to the upper bound in (3.3), we may choose $\delta_{D}$ small enough so
that if $\delta<\delta_{D}$ and $z\in S(0;\delta)$, then
$z+r\hat{\eta}(z)\notin S(0;\delta)$ for any $r>0$. Thus, $z+r\hat{\eta}(z)\in
bC(0;\delta)$ if and only if $|L(z+r\hat{\eta}(z),0)|=\delta$. Therefore, we
must have that

$\displaystyle r_{0}(z;\delta)$ $\displaystyle=$
$\displaystyle\min\left\\{r\geq 0:|z_{d}+r\hat{\eta}_{d}(z)|=\delta\right\\}.$

Thus,
$\delta=|z_{d}+r_{0}(z;\delta)\hat{\eta}_{d}(z)|\leq|z_{d}|+r_{0}(z;\delta)||\hat{\eta}(z)||=|L(z,0)|+r_{0}(z;\delta)$.
This gives the lower bound in (3.7).

For the upper bound, we solve for $r_{0}(z;\delta)$ in the quadratic equation
$|z_{d}+r_{0}(z;\delta)\hat{\eta}_{d}(z)|^{2}=\delta^{2}$ to obtain that

$\displaystyle r_{0}(z;\delta)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\operatorname{Re}(\overline{z_{d}}\hat{\eta}_{d}(z))^{2}+|\hat{\eta}_{d}(z)|^{2}(\delta^{2}-|z_{d}|^{2})}-\operatorname{Re}(\overline{z_{d}}\hat{\eta}_{d}(z))}{|\hat{\eta}_{d}(z)|^{2}}$
$\displaystyle\leq$
$\displaystyle|\hat{\eta}_{d}(z)|^{-1}\sqrt{(\delta^{2}-|z_{d}|^{2})}+|\hat{\eta}_{d}(z)|^{-2}\left(|\operatorname{Re}(\overline{z_{d}}\hat{\eta}_{d}(z))|-\operatorname{Re}(\overline{z_{d}}\hat{\eta}_{d}(z))\right)$
(3.9) $\displaystyle\leq$
$\displaystyle|\hat{\eta}_{d}(z)|^{-1}\sqrt{(\delta^{2}-|z_{d}|^{2})}+2|\hat{\eta}_{d}(z)|^{-2}\max\\{0,(x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d})\\}.$

Now, let $S^{\prime}(\delta):=\\{z\in
S(0;\delta):x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d}>0\\}$. From (3.1)
and (3.2), we have that

(3.10) $\displaystyle x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d}$
$\displaystyle=$ $\displaystyle-
f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})+ex_{d}^{2}+h(z^{\prime},x_{d})$
$\displaystyle\leq$ $\displaystyle-
c_{D}||z^{\prime}||^{2}+ex_{d}^{2}+h(z^{\prime},x_{d}),$

where $h(z^{\prime},x_{d})=O_{D}(||(z^{\prime},x_{d})||^{3})$ as
$||(z^{\prime},x_{d})||\rightarrow 0$. Thus, (3.10) yields that if $e\leq 0$
and $\delta>0$ is sufficiently small, then
$x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d}\leq 0$ on $S(0;\delta)$, i.e.,
$S^{\prime}(\delta)$ is empty. On the other hand, if $e>0$, then by (3.10),
$||z^{\prime}||\leq C_{D}(|x_{d}|)$ on $S^{\prime}(\delta)$ for some $C_{D}>0$
and sufficiently small $\delta>0$. But $\sup\\{||z^{\prime}||^{2},|x_{d}|:z\in
S(0;\delta)\\}=O_{D}(\delta)$ as $\delta\rightarrow 0$. Thus,

(3.11) $\sup\\{||z^{\prime}||,|x_{d}|:z\in
S^{\prime}(\delta)\\}=O_{D}(\delta)\qquad\text{as}\ \delta\rightarrow 0,$

and $\sup\\{\max\\{0,x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d}\\}:z\in
S(0;\delta)\\}=O_{D}({\delta^{2}})$ as $\delta\rightarrow 0$. Furthermore,

$|\hat{\eta}_{d}(z)|^{-1}=\sqrt{\frac{1+||\nabla
f(z^{\prime},x_{d})||^{2}}{1+\partial_{x_{d}}f(z^{\prime},x_{d})^{2}}}=1+O(||z^{\prime},x_{d}||^{2})\
\text{as}\ ||(z^{\prime},x_{d})||\rightarrow 0.$

Thus, $\sup\\{|\hat{\eta}_{d}(z)|^{-1}:z\in S(0;\delta)\\}=1+O_{D}(\delta)$ as
$\delta\rightarrow 0$. Substituting these estimates into (3.9) gives the
desired upper bound, and completes the proof. ∎

Proof of Theorem 3.7. We use Lemmas 3.5 and 3.8 to prove (3.5) and (3.6),
respectively. First, let $t_{D}$ be as in (3.4). By (3.3), we may choose
$\delta_{D}>0$ sufficiently small so that for any fixed $w\in bD$ and
$\delta<g(w)^{-1}\delta_{D}$, every $z\in C(w;g(w)\delta)$ admits a unique
projection onto $bD$, i.e., there is a unique $\zeta=\zeta_{z}\in bD$ and
$t=t_{z}>0$ such that $z=\zeta+t\hat{\eta}(\zeta)$. Moreover, owing to the
continuous dependence of $t_{z}$ on $||z-w||$, we may also assume that
$t<t_{D}$ for all $z\in C(w;g(w)\delta)$. So, we have that
$R_{w}(\boldsymbol{\delta}(w))\leq t_{D}$. Then, by (3.4)

$t\leq C_{D}|L(\zeta+t\hat{\eta}(\zeta),w)|=C_{D}|L(z,w)|\leq
C_{D}g(w)\delta.$

Thus, $t_{D}\leq C_{D}g(w)\delta$, which implies that
$R_{w}(\boldsymbol{\delta}(w))\leq C_{D}g(w)\delta$. Taking supremum in $w\in
bD$, we obtain (3.5).

Next, for (3.6), observe that since

$\displaystyle r_{w}(\zeta;\boldsymbol{\delta}(w))=\begin{cases}-\infty,\
&\text{when}\ \zeta\notin S(w;\boldsymbol{\delta}(w)),\\\ g(w)\delta,\
&\text{when}\ \zeta=w,\end{cases}$

we have that $g(w)\delta\leq
r_{\infty}(\zeta,\boldsymbol{\delta})=\sup\\{r_{w}(\zeta;\boldsymbol{\delta}(w)):w\in
bD,\ \text{such that}\ \zeta\in S(w;\boldsymbol{\delta}(w))\\}.$ By Lemma 3.8,
there exist $C_{D},\delta_{D}>0$ such that, for
$||\boldsymbol{\delta}||_{\infty}<\delta_{D}$,

$\displaystyle r_{w}(\zeta;\boldsymbol{\delta}(w))$ $\displaystyle\leq$
$\displaystyle g(w)\delta+C_{D}||g||_{\infty}\delta^{2}$ $\displaystyle=$
$\displaystyle
g(\zeta)\delta+(g(w)-g(\zeta))\delta+C_{D}||g||_{\infty}\delta^{2}$
$\displaystyle\leq$ $\displaystyle
g(\zeta)\delta+||g||_{\alpha}||\zeta-w||^{\alpha}\delta+C_{D}||g||_{\infty}\delta^{2}.$

Thus, shrinking $\delta_{D}$ if necessary, we have from Lemma 3.5 that

$\displaystyle
r_{w}(\zeta;\boldsymbol{\delta}(w))-g(\zeta)\delta\leq||g||_{\alpha}||\zeta-w||^{\alpha}\delta+C_{D}||g||_{\infty}\delta^{2}\leq||g||_{\alpha}C_{D}|L(z,w)|^{\frac{\alpha}{2}}\delta\leq
C_{D}||g||_{\alpha}^{2}\delta^{1+\frac{\alpha}{2}},$

for $||\boldsymbol{\delta}||_{\infty}<\delta_{D}$. Taking supremum over $w$,
we obtain (3.6). ∎

######  Corollary 3.9.

Let $D$ be as in Definition 3.1 and $\delta>0$. There exist
$C_{D},\delta_{D}>0$ such that for any $w\in bD$ and $\delta<\delta_{D}$,

$C(w;\delta)\subset T_{C_{D}\delta}(S(w;C_{D}\delta)).$

###### Proof.

Let $C_{D},\delta_{D}$ be as in Theorem 3.7 (for
$\boldsymbol{\delta}\equiv\delta$) and $t_{D}$ be as in Lemma 3.5. Shrinking
$\delta_{D}>0$ further, if necessary, we may assume that
$C_{D}\delta_{D}<t_{D}$, and each $z\in C(w;\delta)$ has a unique projection
$\zeta$ onto $bD$ whenever $\delta<\delta_{D}$. Thus,
$z=\zeta+t\hat{\eta}(\zeta)$ for some $t<R_{\infty}(\boldsymbol{\delta})\leq
C_{D}\delta<t_{D}$. Now, owing to (3.4), $|L(z,w)|\lesssim\delta$, $z,w\in
bD$, and the proof is complete. ∎

###  3.3. Area and volume estimates

The depth estimates obtained in the previous subsection give bounds on the
volumes of the unions of caps and tubular caps of an induced polyhedron. We
now obtain more precise area and volume estimates for individual caps, cuts,
and visibility regions of a strongly ${\mathbb{C}}$-convex domain. Such
estimates for model quadratic domains have been obtained in Section C. We now
extend these estimates to general strongly ${\mathbb{C}}$-convex domains via
second-order approximation (see Remark 3.4).

######  Theorem 3.10.

Let $D$ be as in Definition 3.1. Let $\nu_{D}$ be the complex-restricted
curvature of $bD$, $\kappa_{d}$ be the Lebesgue volume of the unit ball in
$\mathbb{R}^{d}$, and
$h_{d}=2\int_{0}^{\pi/2}\cos^{d}\\!\theta\>\text{d}\theta$. Then there exist
$C_{D},\delta_{D}>0$ such that for $w\in bD$, $\delta<\delta_{D}$, and
$\boldsymbol{\delta}\equiv\delta$,

(3.12)
$\displaystyle\left|\sigma\big{(}S(w;\delta)\big{)}-h_{d}k_{d}(w)\,\delta^{d}\right|\leq
C_{D}\delta^{d+\frac{1}{2}},$ (3.13)
$\displaystyle\left|\lambda\big{(}C(w;\delta)\big{)}-\frac{h_{d+1}k_{d}(w)}{d}\,\delta^{d+1}\right|\leq
C_{D}\delta^{d+\frac{5}{4}},$ (3.14) $\displaystyle
0\leq\lambda\big{(}C(w;\delta)\big{)}-\lambda\big{(}TC(w;\delta)\big{)}\leq
C_{D}\delta^{2d+1},$ (3.15)
$\displaystyle\left|\sigma\big{(}G_{t\delta}(w;\boldsymbol{\delta})\big{)}-h_{d}(t)k_{d}(w)\,\delta^{d}\right|\leq
C_{D}\delta^{d+\frac{1}{2}},\ t\in\left[0,1-C_{D}\sqrt{\delta}\right],$

where $k_{d}(w)=4\kappa_{2d-2}\nu_{D}(w)^{-1/2}$ and
$h_{d}(t)=2\int_{0}^{\sin^{-1}\sqrt{1-t^{2}}}(\cos\theta-t)^{d-1}\cos\theta\,d\theta$,
$t\in[0,1]$.

######  Remark 3.11.

While the surface areas of $S(w;\delta)$ and $G_{0}(w;\boldsymbol{\delta})$,
$\boldsymbol{\delta}\equiv\delta$, coincide up to first order in $\delta$
(since $h_{d}(0)=h_{d}$), they are not the same sets unless the quasimetric
$\mathtt{d}$ defined in Section 3.4 is a metric.

###### Proof.

First, we introduce some notation. The map
$\pi_{2d-1}:\mathbb{C}^{d}\rightarrow{\mathbb{C}}^{d-1}\times\mathbb{R}$
denotes the projection
$(z_{1},...,z_{d})\mapsto(z^{\prime},x_{d})=(z_{1},...,z_{d-1},x_{d})$. The
set $\pi_{2d-1}(C)$ is denoted by $\widetilde{C}$ for any
$C\subset\mathbb{C}^{d}$. Whenever needed, the dependence of cuts, tubular
cuts, caps and visibility regions on the underlying domain is be indicated via
a superscript, as in $C^{D}(w;\delta)$, $S^{D}(z;\delta)$, etc.

Let $\varepsilon_{D}>0$ be as in Lemma 3.3. Let $\delta^{*}_{D}>0$ be such
that $B_{w}(\delta^{*}_{D})\subseteq U_{w}^{-1}(B_{0}(\varepsilon_{D}))$ for
all $w\in bD$, where $U_{w}$ is the unitary transformation granted by Lemma
3.3. Owing to the estimates (3.3), (3.4) and (3.6), there is a $\delta_{D}>0$
such that for all $\delta<\delta_{D}$, $t>0$,

$C(w;\delta),S(w;\delta),G_{t\delta}(w;\delta)\subset B_{w}(\delta^{*}_{D}).$

In view of this, we assume that $w=0$ and $D=D^{w}=\\{z\in
B_{0}(\varepsilon_{D}):y_{d}>f_{w}(z^{\prime},x_{d})\\}$.

We denote $f_{w}$ by $f$, and refer the reader to Remark 3.4 for some useful
observations. It is also useful to note that, as
$||(z^{\prime},x_{d})||\rightarrow 0$ and $\delta\rightarrow 0$,

  * $(a)$

$f(z^{\prime},x_{d})=f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})+O_{D}(||x_{d}z^{\prime}||+|x_{d}|^{2}+||(z^{\prime},x_{d})||^{3})$,
where $f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ is as in Remark 3.4,

  * $(b)$

$||\nabla f(z^{\prime},x_{d})||,||\nabla f(z^{\prime},x_{d})-\nabla
f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d})||,|\partial_{x_{d}}f(z^{\prime},x_{d})|=O_{D}(||(z^{\prime},x_{d})||)$,

  * $(c)$

$\dfrac{\text{d}(\pi_{2d-1})_{*}\sigma}{\text{d}\lambda_{2d-1}}(z^{\prime},x_{d})=\left(1+||\nabla
f(z^{\prime},x_{d})||^{2}\right)^{\frac{1}{2}}=1+O_{D}(||(z^{\prime},x_{d})||^{2})$,

  * $(d)$

$\sup_{z\in K_{\delta}}\\{||z^{\prime}||^{2},|x_{d}|\\}=O_{D}(\delta)$, where
$K_{\delta}$ is either $C(0;\delta)$, $S(0;\delta)$, or
$G_{t\delta}(0;\delta)$, $t>0$.

We now note a common theme in this proof. The quantities
$\sigma\left(S(0;\delta)\right)$, $\lambda\left(TC(0;\delta)\right)$, and
$\sigma\left(G_{t\delta}(0;\delta)\right)$ can be expressed as certain
integrals (with respect to $\sigma$) on suitable subsets of $bD$. If these
subsets are small, then due to $(b)$ and $(c)$ above, these integrals can be
expressed as integrals (with respect to $\lambda_{2d-1}$) over the projections
of these subsets onto $\\{y_{d}=0\\}$ without a significant loss, making them
easier to tackle.

The estimates above suggest that it is useful to compute the volumes and areas
of $\delta$-cuts, $\delta$-caps and $(\delta,s)$-visibility regions at $0$ for
model domains of the form

(3.16)
$D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}:=\left\\{(z_{1},...,z_{d})\in\mathbb{C}^{d}:y_{d}>f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})\right\\}.$

These computations are relatively straightforward, and have been placed in
Appendix C.

We first prove (3.12). Since $S(0;\delta)\subset bD^{w}$ and
$y_{d}=f(z^{\prime},x_{d})$ for $z_{d}\in bD^{w}$, $S(0;\delta)$ is the graph
of $f$ over

$\widetilde{S}(0;\delta)=\\{(z^{\prime},x_{d}):x_{d}^{2}+f(z^{\prime},x_{d})^{2}\leq\delta^{2}\\}.$

Similarly, $S^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)$ is
the graph of $f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ over

$\widetilde{S}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)=\\{(z^{\prime},x_{d}):x_{d}^{2}+f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})^{2}\leq\delta^{2}\\}.$

Thus, by $(a)$ and $(d)$ above, there exist $a_{D},\delta_{D}>0$, such that

(3.17)
$\widetilde{S}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta-
a_{D}\delta^{3/2})\subseteq\widetilde{S}(0;\delta)\subseteq\widetilde{S}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta+a_{D}\delta^{3/2}),\quad\delta<\delta_{D}.$

Observe that when $S$ and $f$ are either $S(0;\delta)$ and $f$, respectively,
or $S^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)$ and
$f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$, respectively, then

(3.18)
$\displaystyle|\lambda_{2d-1}(\widetilde{S})-\sigma(S)|=\left|\int_{\widetilde{S}}d\sigma(z)-\int_{S}d\sigma(z)\right|$
$\displaystyle=$ $\displaystyle\int_{\widetilde{S}}\left(\sqrt{1+||\nabla
f(z^{\prime},x_{d})||^{2}}-1\right)d\lambda_{2d-1}$ $\displaystyle=$
$\displaystyle\lambda_{2d-1}(\widetilde{S})O_{D}(\delta)\quad\text{as}\
\delta\rightarrow 0.$

Now, (3.12) follows from (3.17) and (C.2) once we recall from Lemma 3.2 that
$16\nu_{D}(w)=v_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}^{2}$.

Next, we prove (3.13) for $TC$ instead of $C$. Then, once we prove (3.14),
(3.13) will automatically follow. Recall that
$TC(0;\delta)=T_{r_{0}(\cdot;\delta)}(S(0;\delta))$. Now, setting
$n(z)=\sqrt{1+||\nabla f(z^{\prime},x_{d})||^{2}}$ and
$\eta_{d}(z)=-\partial_{x_{d}}f(z^{\prime},x_{d})+i$, we have by (3.2) and
(3.1) that for $z\in S(0;\delta)$,

(3.19) $\displaystyle r_{0}(z;\delta)$ $\displaystyle=$
$\displaystyle|\hat{\eta}_{d}(z)|^{-2}\left(\sqrt{(\operatorname{Re}(\hat{\eta}_{d}(z)\overline{z_{d}}))^{2}+|\hat{\eta}_{d}(z)|^{2}(\delta^{2}-|z_{d}|^{2})}-\operatorname{Re}(\hat{\eta}_{d}(z)\overline{z_{d}})\right)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{(\operatorname{Re}(\eta_{d}(z)\overline{z_{d}}))^{2}+(1+(\partial_{x_{d}}f(z^{\prime},x_{d}))^{2})(\delta^{2}-x_{d}^{2}-y_{d}^{2})}-y_{d}+x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})}{|\hat{\eta}_{d}(z)|^{2}n(z)}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\delta^{2}-x_{d}^{2}+(\delta^{2}-y_{d}^{2})(\partial_{x_{d}}f(z^{\prime},x_{d}))^{2}-2y_{d}x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})}-f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})+O_{D}(\delta^{\frac{3}{2}})}{|\hat{\eta}_{d}(z)|^{2}n(z)}$
$\displaystyle=$
$\displaystyle\sqrt{\delta^{2}-x_{d}^{2}}-f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})+O_{D}(\delta^{5/4}),$

as $\delta\rightarrow 0$. Here, we have used that
$\sqrt{|a|}-\sqrt{|b|}\leq\sqrt{a+b}\leq\sqrt{|a|}+\sqrt{|b|}$, and
$x_{d},y_{d},(\partial_{x_{d}}f)^{2}=O_{D}(\delta)$ on $S(0;\delta)$. Thus, by
Roccaforte’s formula (see Theorem 3.6) and arguing as in the proof of (3.12)
above,

$\displaystyle\lambda\big{(}TC(0;\delta)\big{)}$ $\displaystyle=$
$\displaystyle\int_{\widetilde{S}(0;\delta)}\left(\sqrt{\delta^{2}-x_{d}^{2}}-f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})\right)\,d\lambda_{2d-1}(z^{\prime},x_{d})+O_{D}(\delta^{d+5/4}),$
$\displaystyle=$
$\displaystyle\int_{\widetilde{S}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)}\left(\sqrt{\delta^{2}-x_{d}^{2}}-f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime})\right)\,d\lambda_{2d-1}(z^{\prime},x_{d})+O_{D}(\delta^{d+5/4}),$
$\displaystyle=$
$\displaystyle\lambda\big{(}C^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)\big{)}+O_{D}(\delta^{d+5/4})\qquad\text{as}\
\delta\rightarrow 0.$

Now, invoking Theorem C.1, we have that for some $\delta_{D},C_{D}>0$,

$\left|\lambda\big{(}TC(w;\delta)\big{)}-\frac{h_{d+1}}{2}k_{d}(w)\,\delta^{d+1}\right|\leq
C_{D}\delta^{d+\frac{5}{4}},\qquad\delta<\delta_{D}.$

To prove (3.14), we must consider the portion of $C(0;\delta)$ that is not
contained in $TC(0;\delta)$. By Corollary 3.9, we may assume that
$C_{D},\delta_{D}>0$ are such that $C(0;\delta)\subset
T_{C_{D}\delta}(S(0;C_{D}\delta))$, and each $\zeta\in C(0;\delta)$ has a
unique projection $z$ onto $S(0;C_{D}\delta)$. Thus, $\zeta=z+t\hat{\eta}(z)$
for some $z\in S(0;C_{D}\delta)$ and $t<C_{D}\delta$. Writing
$z=(z^{\prime},x_{d}+iy_{d})$, the condition $z+t\hat{\eta}(z)\in C(0;\delta)$
holds if and only if $t>0$, and $|L(\zeta,0)|<\delta$, i.e.,

(3.20)
$x_{d}^{2}+y_{d}^{2}+\frac{t}{n(z)}(2y_{d}-2x_{d}\partial_{x_{d}}f(z^{\prime},x_{d}))+\frac{t^{2}}{n(z)^{2}}(1+(\partial_{x_{d}}f(z^{\prime},x_{d}))^{2})^{2}<\delta^{2},$

where $n(z)=\sqrt{1+||\nabla f(z^{\prime},x_{d})||^{2}}$. The solution set of
the above pair of inequalities is $(0,r_{0}(z;\delta))$ when
$x_{d}^{2}+y_{d}^{2}\leq\delta^{2}$, i.e., when $z\in S(0;\delta)$. In this
case, $\zeta\in TC(0;\delta)$. On the other hand, if
$x_{d}^{2}+y_{d}^{2}>\delta^{2}$ and $y_{d}\geq
x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})$, the solution set is empty. Thus,

$C(0;\delta)\setminus TC(0;\delta)\subseteq
T_{t_{\max}}\big{(}S^{\prime\prime}(\delta)\big{)},$

where $S^{\prime\prime}(\delta)=\\{z\in S(0;C_{D}\delta)\setminus
S(0;\delta):x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d}>0\\}$ and
$t_{\max}(z)$ is the maximum positive solution of (3.20) for
$z=(z^{\prime},x_{d}+if(z^{\prime},x_{d}))$. As in the proof of (3.11) in
Lemma 3.8, $\sup\\{||(z^{\prime},x_{d})||:(z^{\prime},x_{d})\in
S^{\prime\prime}(\delta)\\}=O_{D}(\delta)$ as $\delta\rightarrow 0$. Thus,
$\lambda_{2d-1}(S^{\prime\prime}(\delta))=O_{D}(\delta^{2d-1})$ Also, since

$t_{\max}(z)\leq\frac{(x_{d}\partial_{x_{d}}f(z^{\prime},x_{d})-y_{d})n(z)}{(1+(\partial_{x_{d}}f(z^{\prime},x_{d}))^{2})^{2}}\qquad\text{when}\
z\in S^{\prime\prime}(\delta),$

$\sup_{S^{\prime\prime}(\delta)}(t_{\max})=O_{D}(\delta^{2})$. Thus,

$\displaystyle\lambda\big{(}C(0;\delta)\setminus
TC(0;\delta)\big{)}=O_{D}(\sigma(S^{\prime\prime}(\delta))\cdot\delta^{2})$
$\displaystyle=O_{D}(\lambda_{2d-1}(S^{\prime\prime}(\delta))\cdot\delta^{2})=O_{D}(\delta^{2d+1})$

as $\delta\rightarrow 0$. This completes the proof of (3.14).

Lastly, we prove (3.15). Note that $G_{t\delta}(0;\delta)=\\{z\in
bD:|L(0+t\delta\hat{\eta}(0),z)|<\delta\\}$ is the graph of $f$ over

$\displaystyle\widetilde{G}_{t\delta}(0;\delta)$ $\displaystyle=$
$\displaystyle\left\\{(z^{\prime},x_{d}):\left(t\delta+q_{1}(z^{\prime},x_{d})\right)^{2}+\left(x_{d}+q_{2}(z^{\prime},x_{d})\right)^{2}<\delta^{2}n(z)^{2}\right\\},$

where

$\displaystyle q_{1}(z^{\prime},x_{d})$ $\displaystyle=$ $\displaystyle
2\sum_{j=1}^{d-1}\operatorname{Re}(z_{j}\partial_{z_{j}}f(z^{\prime},x_{d}))-f(z^{\prime},x_{d})+x_{d}\partial_{x_{d}}f(z^{\prime},x_{d}),$
$\displaystyle q_{2}(z^{\prime},x_{d})$ $\displaystyle=$ $\displaystyle
2\sum_{j=1}^{d-1}\operatorname{Im}(z_{j}\partial_{z_{j}}f(z^{\prime},x_{d}))-\partial_{x_{d}}f(z^{\prime},x_{d})(t\delta-f(z^{\prime},x_{d})).$

Similarly,
$G^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}_{t\delta}(0;\delta)$ is
the graph of $f_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}$ over

$\widetilde{G}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}_{t\delta}(0;\delta)=\left\\{(z^{\prime},x_{d}):(t\delta+q_{1}^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d}))^{2}+(x_{d}+q_{2}^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d}))^{2}<\delta^{2}n^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z)^{2}\right\\},$

where
$q_{1}^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d})=\sum_{j=1}^{d-1}\left(\alpha_{j}|z_{j}|^{2}+\beta_{j}\operatorname{Re}\overline{z}_{j}^{2}\right)$,
$q_{2}^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d})=2\sum_{j=1}^{d-1}\beta_{j}\operatorname{Im}z_{j}^{2}$,
and
$n^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d})=\sqrt{1+4\sum_{j=1}^{d-1}|\alpha_{j}z_{j}+\beta_{j}\overline{z}_{j}|^{2}}$.
Now, from $(a)$, we have that

$q_{j}(z^{\prime},x_{d})=q_{j}^{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}(z^{\prime},x_{d})+O_{D}(||(z^{\prime},x_{d})||),\quad
j=1,2.$

Thus, owing to $(d)$, there exist $a_{D},\delta_{D}>0$ such that

(3.21)
$\widetilde{G}_{t\delta}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta(1-a_{D}\delta^{\frac{1}{2}}))\subseteq\widetilde{G}_{t\delta}(0;\delta)\subseteq\widetilde{G}_{t\delta}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta(1+a_{D}\delta^{\frac{1}{2}})),\quad
0\leq t\leq 1-a_{D}\delta^{\frac{1}{2}},$

for all $\delta<\delta_{D}$. Once again, as in (3.18), we have that

$|\lambda_{2d-1}(\widetilde{G})-\sigma(G)|=\lambda_{2d-1}(\widetilde{G})O_{D}(\delta)\quad\text{as}\
\delta\rightarrow 0$

for both $G=G_{t\delta}(0;\delta)$ and
$G=G_{t\delta}^{D_{{\boldsymbol{\alpha}},{\boldsymbol{\beta}}}}(0;\delta)$,
$t<1$. Moreover, since $h_{d}$ is Lipschitz on $[0,1]$,

$\left|h_{d}\left(\frac{t}{1+O_{D}(\delta^{\frac{1}{2}})}\right)-h_{d}\left(t\right)\right|=O_{D}(\sqrt{\delta})\quad\text{as}\
\delta\rightarrow 0,$

with constants independent of $t\in[0,1]$. Thus, the claim now follows from
(3.21) and Theorem C.2. ∎

Note that the estimate (3.15) has been obtained under the assumption that the
depth function $\boldsymbol{\delta}$ is a constant function. A more general
estimate can be deduced from this special case.

######  Corollary 3.12.

Let $D$, $\nu_{D}$, $\kappa_{d}$, $h_{d}(t)$, and $k_{d}(w)$ be as in Theorem
3.10. Let $\boldsymbol{\delta}=g\delta$ for some positive function
$g\in\mathcal{C}^{\alpha}(bD)$, $\alpha\in(0,1)$, and $\delta>0$. Then there
exist $C_{D},\delta_{D}>0$ such that for $w\in bD$ and
$||\boldsymbol{\delta}||_{\infty}<\delta_{D}$,

(3.22)
$\displaystyle\qquad\left|\sigma\big{(}G_{t\boldsymbol{\delta}(w)}(w;\boldsymbol{\delta})\big{)}-h_{d}(t)k_{d}(w)g(w)^{d}\,\delta^{d}\right|\leq
C_{D}||g||_{\alpha}^{d+1}\delta^{d+\frac{\alpha}{2}},\
t\in\left[0,1-C_{D}\sqrt{\delta}\right].$

###### Proof.

First, as in the proof of (3.6), there exist $C_{D},\delta_{D}>0$ such that
for any $w\in bD$, $||\boldsymbol{\delta}||_{\infty}<\delta$, and $\zeta\in
G_{t\boldsymbol{\delta}(w)}(w;\boldsymbol{\delta})$,

$0<g(w)-C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}}\leq g(\zeta)\leq
g(w)+C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}}.$

Now, for a fixed $w\in bD$, let $\boldsymbol{\delta}_{1}\equiv\delta_{1}$ and
$\boldsymbol{\delta}_{2}\equiv\delta_{2}$ be constant functions on $bD$ given
by

$\displaystyle\delta_{1}$ $\displaystyle:=$ $\displaystyle
g(w)\delta\left(1-C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}}\right),$
$\displaystyle\delta_{2}$ $\displaystyle:=$ $\displaystyle
g(w)\delta\left(1+C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}}\right).$

Then,

$G_{1}:=G_{t_{1}\boldsymbol{\delta}_{1}(w)}(w;\boldsymbol{\delta}_{1})\subseteq
G_{t\boldsymbol{\delta}(w)}(w;\boldsymbol{\delta})\subseteq
G_{t_{2}\boldsymbol{\delta}_{2}(w)}(w;\boldsymbol{\delta}_{2})=:G_{2},$

where $t_{1}=t(1-C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}})^{-1}$ and
$t_{2}=t(1+C_{D}||g||_{\alpha}^{2}\delta^{\frac{\alpha}{2}})^{-1}$. Since
$t\in[0,1-C_{D}\sqrt{\delta}]$, shrinking $\delta_{D}>0$ if necessary, we
obtain that $t_{j}\in[0,1-C_{D}\sqrt{\delta_{j}}]$ for $j=1,2$. Thus, the
estimate (3.15) applies to both $G_{1}$ and $G_{2}$, and we obtain (3.22). ∎

###  3.4. Containment and coverage

In this section, we express the containment condition on an induced polyhedron
$P$ equivalently as a purely boundary-based coverage condition on the caps of
$P$. In [22], the authors consider a natural quasimetric on the boundary of a
strongly ${\mathbb{C}}$-convex domain. We consider an appropriate modification
of their construction. The balls in this quasimetric coincide with caps of
induced polyhedra. This allows us to convert the containment condition on an
induced polyhedron into a covering condition on $bD$.

######  Lemma 3.13 ([22, Proposition 3.3]).

Let $D$ and $L$ be as in Definition 3.1. The function $\mathtt{d}:bD\times
bD\rightarrow[0,\infty)$ given by

$\mathtt{d}(\zeta,w)=|L(\zeta,w)|^{\frac{1}{2}}$

is a quasimetric on $bD$, i.e., there is a $\mathfrak{q}_{D}\geq 1$ such that

  1. $(a)$

$\mathtt{d}(\zeta,w)=0$ if and only if $\zeta=w$,

  2. $(b)$

(quasi-symmetry)
$\mathtt{d}(\zeta,w)\leq\mathfrak{q}_{D}\,\mathtt{d}(w,\zeta)$ for all
$\zeta,w\in bD$, and

  3. $(c)$

(quasi-triangle inequality)
$\mathtt{d}(\zeta,w)\leq\mathfrak{q}_{D}(\mathtt{d}(\zeta,z)+\mathtt{d}(z,w))$,
for all $z,w,\zeta\in bD$.

###### Proof.

Property $(a)$ follows from (3.3). To obtain $(b)$, note that by the
$\mathcal{C}^{2}$-smoothness of $\rho$,

$|\mathtt{d}(\zeta,w)^{2}-\mathtt{d}(w,\zeta)^{2}|\leq\big{|}|L(\zeta,w)|-|L(w,\zeta)|\big{|}=\left|\left<\frac{\partial\rho(w)}{||\partial\rho(w)||}-\frac{\partial\rho(\zeta)}{||\partial\rho(\zeta)||},\zeta-w\right>\right|\lesssim||\zeta-w||^{2}.$

Now, the claim follows from (3.3). Finally, for claim $(c)$, observe that

$\displaystyle\mathtt{d}^{2}(\zeta,w)-\mathtt{d}^{2}(z,w)-\mathtt{d}^{2}(\zeta,z)$
$\displaystyle\leq\left|\left<\frac{\partial\rho(w)}{||\partial\rho(w)||},w-\zeta\right>-\left<\frac{\partial\rho(w)}{||\partial\rho(w)||},w-z\right>-\left<\frac{\partial\rho(z)}{||\partial\rho(z)||},z-\zeta\right>\right|$
$\displaystyle\lesssim\left|\left<\partial\rho(w)-\partial\rho(z),z-\zeta\right>\right|$
$\displaystyle\lesssim||w-z||\times||z-\zeta||$
$\displaystyle\lesssim\mathtt{d}(z,w)\mathtt{d}(\zeta,z),$

where the final two steps follow from the $\mathcal{C}^{2}$-smoothness of
$\rho$ and (3.3), respectively. Thus, we obtain $(c)$. ∎

We now observe that the containment condition $P(\varphi;\delta)\subset D$ can
indeed be reduced to a covering condition on $bD$ in terms of the caps of
$P(\varphi;\delta)$, which are $\mathtt{d}$-balls since

$S(w;\delta):=\\{\zeta\in bD:\mathtt{d}(\zeta,w)\leq\sqrt{\delta}\\}.$

######  Lemma 3.14.

Let $D$ be as in Definition 3.1, $\varphi=\\{w^{1},...,w^{n}\\}\subset bD$,
and $\boldsymbol{\delta}:bD\rightarrow(0,\infty)$. Then,
$P(\varphi;\boldsymbol{\delta})$ is contained in $D$ if and only if
$bD=S(\varphi;\boldsymbol{\delta})$, where

$S(\varphi;\boldsymbol{\delta}):=\bigcup_{j=1}^{n}S(w^{j};\boldsymbol{\delta}(w^{j})).$

###### Proof.

Recall that $P(\varphi;\boldsymbol{\delta})$ is the union of those connected
components of

$\widetilde{P}(\varphi;\boldsymbol{\delta})=\bigcap_{j=1}^{n}\left\\{z\in\mathbb{C}^{d}:\left|L(z,w^{j})\right|>\boldsymbol{\delta}(w^{j})\right\\}$

that intersect $D$. Since
$S(\varphi;\boldsymbol{\delta})=bD\setminus\widetilde{P}(\varphi;\boldsymbol{\delta})$,
$bD\neq S(\varphi;\boldsymbol{\delta})$ if and only if
$bD\cap\widetilde{P}(\varphi;\boldsymbol{\delta})$ is nonempty. However, if
$z\in bD\cap\widetilde{P}(\varphi;\boldsymbol{\delta})$, then $z$ must be
contained in a connected component of
$\widetilde{P}(\varphi;\boldsymbol{\delta})$ that intersects $D$, i.e.,
$bD\cap\widetilde{P}(\varphi;\boldsymbol{\delta})=bD\cap
P(\varphi;\boldsymbol{\delta})$. Thus, it suffices to show that
$P(\varphi;\boldsymbol{\delta})\subset D$ if and only if $bD\cap
P(\varphi;\boldsymbol{\delta})$ is empty. The forward implication is trivial.
For the converse, we argue by contraposition: if $P$ is a connected component
of $P(\varphi;\delta)$ that is not contained in $D$, then it must intersect
$bD$. Otherwise $P$ is the disjoint union of the nonempty open sets $P\cap D$
and $P\cap(\mathbb{C}^{d}\setminus\overline{D})$ which is impossible. ∎

In order to study the covering problem on $bD$, we need the existence of a
‘good’ cover of $bD$. The existence of this cover is due to the fact that
balls in the quasimetric $\mathtt{d}_{D}$ satisfy a Vitali-type covering
lemma; see [20, Lemma 4.1.1] for a proof that generalizes to quasimetrics.

######  Lemma 3.15.

Let $D$ be as in Definition 3.1. There is a constant
$0<\mathfrak{K}=\mathfrak{K}_{D}\leq 1/9$ such that for any $\delta>0$, there
exist $m=m_{\delta}\in{\mathbb{N}}$ and $p^{1},...,p^{m}\in bD$ such that

  * $(a)$

$bD=S(p^{1};\delta)\cup\cdots\cup S(p^{m};\delta)$, and

  * $(b)$

$S(p^{1},\mathfrak{K}\>\delta),...,S(p^{m};\mathfrak{K}\>\delta)$ are pairwise
disjoint.

The dilation constant in the above lemma depends on the quasimetric constant
$\mathfrak{q}_{D}$. In the case of the unit ball $\mathbb{B}^{d}$,
$\mathtt{d}$ is in fact a metric on $b\mathbb{B}^{d}$ (e.g., see [37, Section
5.1]), so we may choose $\mathfrak{q}_{\mathbb{B}^{d}}$ and
$\mathfrak{K}_{\mathbb{B}^{d}}$ as $1$ and $1/9$, respectively.

###  3.5. Nearest-neighbor estimates

So far, we have focussed on estimates for individual cuts and caps, but for
variance bounds and normal approximation results, it is crucial to understand
volume and area contributions of intersecting cuts and caps. In this
subsection, we collect a couple of such estimates.

First, we quantify the following statement: cuts based at points that are
sufficiently far away from each other (in $\mathtt{d}$) cannot intersect.

######  Proposition 3.16.

Let $D$ be as in Definition 3.1. There are constants $\delta_{D},A_{D}>0$ such
that

$\sigma\\{\zeta\in bD:C(\zeta;\delta)\cap C(w;\delta)\neq\emptyset\\}\leq
A_{D}\delta^{d}\qquad\forall w\in bD,\ \delta<\delta_{D}.$

###### Proof.

Let $C_{D},\delta_{D}$ be as in Corollary 3.9. Let $w,\zeta\in bD$ be such
that $C(w;\delta)\cap C(\zeta;\delta)\neq\emptyset$ for some
$\delta<\delta_{D}$. Then, $T_{C_{D}\delta}(S(w;C_{D}\delta))\cap
T_{C_{D}\delta}(S(\zeta;C_{D}\delta))\neq\emptyset$. We may assume that each
$z\in T_{C_{D}\delta}(bD)$ admits a unique projection onto $bD$. Then, there
is a $z\in S(w;C_{D}\delta)\cap S(\zeta;C_{D}\delta)$. By the quasi-symmetry
of $\mathtt{d}$, $\zeta\in S(w;C_{D}\delta)$, and the claim follows from
(3.12). ∎

Next, we bound (from below) the volume increment caused by adding a cut at a
point that is sufficiently isolated from the rest of the source points. Note
that this result is stated for tubular cuts, which suffices for our purposes.

######  Proposition 3.17.

Let $D$ be as in Definition 3.1. Let $\boldsymbol{\delta}=g\delta$ for some
positive function $g\in\mathcal{C}^{\alpha}(bD)$, $\alpha\in(0,1)$, and
$\delta>0$. There exist $b_{D},B_{g,D},t_{g,D}>0$ such that for all $w\in bD$
and $t<t_{g,D}$,

$\lambda\left(TC(w;\boldsymbol{\delta}(w))\setminus\bigcup_{\zeta\in
bD\setminus
S(w;t\boldsymbol{\delta}(w))}TC(\zeta;\boldsymbol{\delta}(\zeta))\right)\geq
b_{D}\left(t^{2}\boldsymbol{\delta}(w)\right)^{d+1},\quad\delta<B_{g,D}t^{4/\alpha}.$

###### Proof.

Let $C_{D}$ and $\delta_{D}$ be as in the proof of Proposition 3.16. Observe
that for any $w\in bD$,

$\\{\zeta\in bD:TC(w;\boldsymbol{\delta}(w))\cap
TC(\zeta;\boldsymbol{\delta}(\zeta))\neq\emptyset\\}\subset\\{\zeta\in
bD:C(w;||g||_{\infty}\delta)\cap
C(\zeta;||g||_{\infty}\delta)\neq\emptyset\\},$

and the latter set is contained in $S(w;C_{g,D}\delta)$, as shown in the proof
of Proposition 3.16. Thus, it suffices to work with $\zeta\in
S(w;C_{g,D}\delta)\setminus S(w;t\boldsymbol{\delta}(w))$ to establish the
claim.

We now bound from below
$r_{w}(z;\boldsymbol{\delta}(w))-r_{\zeta}(z;\boldsymbol{\delta}(\zeta))$ for
certain $z$ and $\zeta$ near $w$. Paraphrasing (3.7) in terms of the
quasimetric $\mathtt{d}$, we have that for $w\in bD$, $z\in
S(w;\boldsymbol{\delta}(w))$, and $\delta<\delta_{g,D}$,

(3.23) $\boldsymbol{\delta}(w)-\mathtt{d}(z,w)^{2}\leq
r_{w}(z;\boldsymbol{\delta}(w))\leq\sqrt{\boldsymbol{\delta}(w)^{2}-\mathtt{d}(z,w)^{4}}+C_{g,D}\delta^{2}.$

Fix $s,t>0$ such that $s<t<1$. Then, if $z\in S(w;s\boldsymbol{\delta}(w))$
and $\zeta\in bD\setminus S(w;t\boldsymbol{\delta}(w))$,

$\sqrt{t\boldsymbol{\delta}(w)}\leq\mathtt{d}(\zeta,w)\leq\mathfrak{q}^{2}\mathtt{d}(z,\zeta)+\mathfrak{q}\mathtt{d}(z,w)\leq\mathfrak{q}^{2}\mathtt{d}(z,\zeta)+\mathfrak{q}^{2}\sqrt{s\boldsymbol{\delta}(w)},$

where $\mathfrak{q}=\mathfrak{q}_{D}\geq 1$ is the quasi-metric constant in
Lemma 3.13. Thus,

$\mathtt{d}(z,\zeta)\geq\left(\mathfrak{q}^{-2}\sqrt{t}-\sqrt{s}\right)\sqrt{\boldsymbol{\delta}(w)}>0,\qquad\text{for}\
s<t/\mathfrak{q}^{4}.$

Now, setting $s=t^{2}/4\mathfrak{q}^{8}$, it follows from (3.23) and the lower
bound on $\mathtt{d}(z,\zeta)$ that for $\zeta\in S(w;C_{g,D}\delta)\setminus
S(w;t\boldsymbol{\delta}(w))$ and $z\in S(w;s\boldsymbol{\delta}(w))\cap
S(\zeta;\boldsymbol{\delta}(\zeta))$,

(3.24)
$r_{w}(z;\boldsymbol{\delta}(w))\geq\boldsymbol{\delta}(w)R(t)\qquad\text{and}\qquad
r_{\zeta}(z;\boldsymbol{\delta}(\zeta))\leq\boldsymbol{\delta}(w)r_{\zeta}(t),$

where $R(t)=1-t^{2}/4\mathfrak{q}^{8}$ and
$r_{\zeta}(t)=\sqrt{\frac{g(\zeta)^{2}}{g(w)^{2}}-\left(\frac{\sqrt{t}}{\mathfrak{q}^{2}}-\frac{t}{2\mathfrak{q}^{4}}\right)^{4}}+C_{g,D}\delta$.

Now, by the assumption on $g$, we obtain that $|g(\zeta)-g(w)|\leq
C_{g,D}\delta^{\frac{\alpha}{2}}$, for all $w\in bD$, $\zeta\in
S(w;C_{g,D}\delta)$, and $\delta<\delta_{g,D}$. Thus, for $w\in bD,\zeta\in
S(w;C_{g,D}\delta)$, and $\delta<\delta_{g,D}$,

$g(\zeta)^{2}\leq g(w)^{2}+C_{g,D}\delta^{\frac{\alpha}{2}}.$

Therefore,

(3.25) $\displaystyle r_{\zeta}(t)$ $\displaystyle\leq$
$\displaystyle\sqrt{1+C_{g,D}\delta^{\frac{\alpha}{2}}-\left(\frac{\sqrt{t}}{\mathfrak{q}^{2}}-\frac{t}{2\mathfrak{q}^{4}}\right)^{4}}+C_{g,D}\delta$
$\displaystyle\leq$ $\displaystyle
1-\frac{1}{2}\left(\frac{\sqrt{t}}{\mathfrak{q}^{2}}-\frac{t}{2\mathfrak{q}^{4}}\right)^{4}+C_{g,D}\delta^{\frac{\alpha}{2}}=:r(t),$

where we have used that $\sqrt{1-b+a}\leq 1-\frac{b}{2}+a$, whenever $b<3/4$
and $a>0$. Now, for sufficiently small $t$,

$\displaystyle
R(t)-r(t)=\left(\frac{(\sqrt{t}-t/2\mathfrak{q}^{2})^{4}}{2\mathfrak{q}^{8}}-\frac{t^{2}}{4\mathfrak{q}^{8}}-C_{g,D}\delta^{\frac{\alpha}{2}}\right)\geq\left(\frac{t^{2}}{8\mathfrak{q}^{8}}-C_{g,D}\delta^{\frac{\alpha}{2}}\right).$

Thus, there are $a_{D},A_{g,D},t_{g,D}>0$ so that $R(t)-r(t)\geq
a_{D}t^{2}\boldsymbol{\delta}(w)$ for all $\delta<A_{g,D}t^{4/\alpha}$,
$t<t_{g,D}$.

Now, let $\pi_{bD}(\xi)$ be the (uniquely determined) projection of $\xi$ onto
$bD$ for $\xi$ sufficiently close to $bD$. Define
$T:=T_{R(t)\boldsymbol{\delta}(w)}(S(w;s\boldsymbol{\delta}(w)))\setminus
T_{r(t)\boldsymbol{\delta}(w)}(S(w;s\boldsymbol{\delta}(w)))$, where
$T_{*}(\cdot)$ is as in Definition 3.1. By (3.24), we have that

$\displaystyle
T_{R(t)\boldsymbol{\delta}(w)}\big{(}S(w;s\boldsymbol{\delta}(w))\big{)}$
$\displaystyle\subset$ $\displaystyle TC(w;\boldsymbol{\delta}(w)),$
$\displaystyle
T_{r(t)\boldsymbol{\delta}(w)}\big{(}S(w;s\boldsymbol{\delta}(w))\big{)}$
$\displaystyle\supset$ $\displaystyle\bigcup_{\zeta\in bD\setminus
S(w;t\boldsymbol{\delta}(w))}\left\\{\xi\in
TC(\zeta;\boldsymbol{\delta}(\zeta)):\pi_{bD}(\xi)\in
S(w;s\boldsymbol{\delta}(w))\right\\}.$

Thus, $T\subset TC(w;\boldsymbol{\delta}(w))\setminus\bigcup_{\zeta\in
bD\setminus
S(w;t\boldsymbol{\delta}(w))}TC(\zeta;\boldsymbol{\delta}(\zeta))$, and it
suffices to produce a lower bound for $\lambda(T)$. We will use Roccaforte’s
formula, i.e., Theorem 3.6 for this.

From (3.12),
$\sigma(S(w;s\boldsymbol{\delta}(w)))\geq\widetilde{a}_{D}(t^{2}\boldsymbol{\delta}(w))^{d}$
for some $\widetilde{a}_{D}>0$. Moreover, $|R(t)|,|r(t)|\leq 2$ for all
$t<t_{g,D}$, and the functions $s_{1},...,s_{2d-1}$ as in Theorem 3.6 are
bounded on $bD$. Thus, by Theorem 3.6 and the above estimates, there are
constants $b_{D},B_{g,D}>0$ so that

$\displaystyle\lambda(T)$ $\displaystyle=$
$\displaystyle\boldsymbol{\delta}(w)(R(t)-r(t))\int\limits_{S(w;s\delta)}\left(1-\sum_{j=1}^{d-1}\frac{(-\boldsymbol{\delta}(w))^{j+1}}{j+1}\frac{R(t)^{j+1}-r(t)^{j+1}}{R(t)-r(t)}s_{j}(z)\right)d\sigma(z)$
$\displaystyle\geq$ $\displaystyle
b_{D}\boldsymbol{\delta}(w)(R(t)-r(t))\sigma(S(w;s\boldsymbol{\delta}(w)))\geq
b_{D}(t^{2}\boldsymbol{\delta}(w))^{d+1}$

for all $\delta<B_{g,D}t^{4/\alpha}$ and $t<t_{g,D}$. ∎

##  4\. Proofs of limit theorems

In this section, we present the proofs of the main results of this paper,
namely those stated in Section 2.2. We first collect some useful observations
regarding the notions introduced in Definition 3.1. We use the notation
introduced in Definition 3.1.

Let $D\subset\mathbb{C}^{d}$ be a strongly ${\mathbb{C}}$-convex
$\mathcal{C}^{2}$-domain as in Definition 3.1,
$\varphi=\\{w^{1},\ldots,w^{m}\\}\subset bD$ be a source set, and
$\boldsymbol{\delta}:bD\to(0,\infty)$ be a size function. Since $bD$ and
$\\{z\in\mathbb{C}^{d}:|L(z,w)|=\delta\\}$, $w\in bD$, $\delta>0$, are smooth
real hypersurfaces in $\mathbb{C}^{d}$, they have zero Lebesgue volume. Thus,

(4.1) $\lambda(D\setminus
P(\varphi;\boldsymbol{\delta}))=\lambda\left(\overline{D}\setminus
P(\varphi;\boldsymbol{\delta})\right)=\lambda\left(D\setminus\overline{P(\varphi;\boldsymbol{\delta})}\right).$

Note that

(4.2)
$D\setminus\overline{P(\varphi;\boldsymbol{\delta})}=\bigcup_{j=1}^{m}\left\\{z\in
D:|L(z,w^{j})|<\boldsymbol{\delta}(w^{j})\right\\}=\bigcup_{j=1}^{m}C(w^{j};\boldsymbol{\delta}(w^{j})).$

On the other hand, recalling that $\overline{C(w;\delta)}\supset
TC(w;\delta)$, we have that

(4.3) $\overline{D}\setminus
P(\varphi;\boldsymbol{\delta})=\bigcup_{j=1}^{m}\left\\{z\in\overline{D}:|L(z,w^{j})|\leq\boldsymbol{\delta}(w^{j})\right\\}\supset\bigcup_{j=1}^{m}TC(w^{j};\boldsymbol{\delta}(w^{j})).$

Setting

$C(\varphi;\boldsymbol{\delta}):=\bigcup_{j=1}^{m}C(w^{j};\boldsymbol{\delta}(w^{j}))\qquad\text{and}\qquad
TC(\varphi;\boldsymbol{\delta}):=\bigcup_{j=1}^{m}TC(w^{j};\boldsymbol{\delta}(w^{j})),$

we obtain from (4.1), (4.3), and (4.2) that

(4.4)
$\lambda\left(TC(\varphi;\boldsymbol{\delta})\right)\leq\lambda\left(D\setminus
P(\varphi;\boldsymbol{\delta})\right)=\lambda\left(C(\varphi;\boldsymbol{\delta})\right)=\lambda\left(\overline{C(\varphi;\boldsymbol{\delta})}\right).$

In order to compute $\lambda\left(TC(\varphi;\boldsymbol{\delta})\right)$
using the tube formula in Theorem 3.6, we express
$TC(\varphi;\boldsymbol{\delta})$ as a $g$-tube of $bD$ for some non-negative
$g$. For this, set $r^{+}(z)=\max\\{r(z),0\\}$, where $r(z):=\max_{1\leq j\leq
m}r_{w^{j}}(z;\boldsymbol{\delta}(w^{j}))$. Then,
$T_{r}=TC(\varphi;\boldsymbol{\delta})$ and
$T_{r^{+}}=TC(\varphi;\boldsymbol{\delta})\cup bD$. Thus,

(4.5)
$\lambda(TC(\varphi;\boldsymbol{\delta}))=\lambda(T_{r})=\lambda(T_{r^{+}}).$

Finally, we note that (4.4) helps estimate
$\delta_{V}(P(\varphi;\boldsymbol{\delta}),D)$ in terms of volumes of cuts
since

(4.6)
$\displaystyle\delta_{V}(P(\varphi;\boldsymbol{\delta}),D)-\lambda(D\setminus
P(\varphi;\boldsymbol{\delta}))$ $\displaystyle=$
$\displaystyle\lambda(P(\varphi;\boldsymbol{\delta})\cap
D)\mathbf{1}[P(\varphi;\boldsymbol{\delta})\not\subset D]$ $\displaystyle\leq$
$\displaystyle\lambda(D)\mathbf{1}[P(\varphi;\boldsymbol{\delta})\not\subset
D].$

In all the proofs below, we first handle the case when the source set of the
random polyhedron is $\eta_{n}$, a Poission point process. Subsequently, the
changes required to tackle the case of the source set being $\beta_{n}$, a
binomial point process, is discussed. For brevity, we set

$\delta_{V}(n):=\delta_{V}\left(D,P(\varphi_{n},\boldsymbol{\delta}_{n})\right),$

where $\varphi_{n}$ is either $\eta_{n}$ or $\beta_{n}$, depending on the
context. We also denote the probability measure $f\sigma$ by $\sigma_{f}$.

###  4.1. Theorem 2.3: convergence in probability

###### Proof of Theorem 2.3.

Let $\eta_{n}$ be a Poisson point process on $bD$ such that (2.5) holds, i.e.,

$\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\subset D)\rightarrow 1\
\text{as}\ n\rightarrow\infty.$

Step 1. Approximating by tubular cuts. First, we observe that by the triangle
inequality

$\displaystyle\left|\frac{\delta_{V}(n)}{\delta_{n}}-\int_{bD}g(z)d\sigma(z)\right|\leq\left|\frac{\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|+\left|\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|$
(4.7)
$\displaystyle+\left|\frac{\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}-\int_{bD}g(z)d\sigma(z)\right|.$

Now, from (4.4) and (4.6), we have that for any $\epsilon>0$,

$\displaystyle\mathbb{P}\left(\left|\frac{\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|>\epsilon\right)=\mathbb{P}\left(\frac{\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}>\epsilon\right)$
$\displaystyle\leq\mathbb{P}\left(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D\right).$

Thus, by the containment assumption on $P(\eta_{n};\boldsymbol{\delta}_{n})$,

(4.8)
$\left|\frac{\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|\stackrel{{\scriptstyle
p}}{{\to}}0\ \text{as}\ n\rightarrow\infty.$

Next, invoking (4.4) and (3.14), we have that

(4.9)
$0\leq\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))\leq\sum_{i=1}^{|\eta_{n}|}\lambda\big{(}C(W^{i};\boldsymbol{\delta}_{n})\setminus
TC(W^{i};\boldsymbol{\delta}_{n})\big{)}\leq
C_{D}|\eta_{n}|\|\boldsymbol{\delta}_{n}\|_{\infty}^{2d+1},$

for some $C_{D}>0$. Thus, taking expectation on both sides, we have that

(4.10)
$\mathbb{E}\left|\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|\leq
C_{D}\|g\|_{\infty}^{2d}n\delta_{n}^{2d}=C_{D}\|g\|_{\infty}^{2d}\frac{(\log
n)^{2}}{n}\to 0$

as $n\to\infty$. Thus, by Markov’s inequality, (4.1), (4.8), and (4.10), (2.6)
follows if

(4.11)
$\mathbb{E}\left|\frac{\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}-\int_{bD}g(z)\text{d}\sigma(z)\right|\rightarrow
0\ \text{as}\ n\rightarrow\infty.$

This reduces to estimating the expectation of a depth function associated to
$P(\eta_{n};\boldsymbol{\delta}_{n})$.

Step 2. Replacing volume by depth. For $z\in bD$, let $r_{n}(z):=\max_{1\leq
j\leq|\eta_{n}|}r_{W^{j}}(z;\boldsymbol{\delta}_{n}(W^{j}))$ and
$r_{n}^{+}(z)=\max\\{r_{n}(z),0\\}.$ Then, by (4.5) the statement (4.11) is
equivalent to

(4.12)
$\mathbb{E}\left|\frac{\lambda(T_{r^{+}_{n}})}{\delta_{n}}-\int_{bD}g(z)\text{d}\sigma(z)\right|\to
0\ \text{as}\ n\rightarrow\infty.$

Using the tube formula in Theorem 3.6 we have that

(4.13)
$\lambda(T_{r^{+}_{n}})=\sum_{j=0}^{d-1}\frac{(-1)^{j}}{j+1}\int_{bD}r^{+}_{n}(z)^{j+1}s_{j}(z)\,\text{d}\sigma(z).$

Now, recalling that $r_{n}^{+}(z)\leq r_{\infty}(z;\boldsymbol{\delta}_{n})$
for all $z\in bD$, and that $r_{\infty}(z;\boldsymbol{\delta}_{n})\leq
C\delta_{n}$ (see (3.6)), we have that for any $j\geq 1$,

$\frac{1}{\delta_{n}}\left|\int_{bD}r^{+}_{n}(z)^{j+1}s_{j}(z)\text{d}\sigma(z)\right|\leq\int_{bD}\frac{r_{\infty}(z;\boldsymbol{\delta}_{n})^{j+1}}{\delta_{n}}|s_{j}(z)|\,\text{d}\sigma(z)\to
0\ \text{as}\ n\rightarrow\infty.$

Thus, since $s_{0}\equiv 1$, to prove (4.12), it suffices to show that

(4.14)
$\mathbb{E}\left|\frac{1}{\delta_{n}}\int_{bD}r^{+}_{n}(z)\,\text{d}\sigma(z)-\int_{bD}g(z)\text{d}\sigma(z)\right|\to
0\ \text{as}\ n\rightarrow\infty.$

Now, by (3.6), $r^{+}_{n}(z)\leq r_{\infty}(z;\boldsymbol{\delta}_{n})\leq
g(z)\delta_{n}+C_{D}\parallel
g\parallel_{\alpha}^{2}\delta_{n}^{1+\frac{\alpha}{2}}$ for some $C_{D}>0$.
Therefore,

$\displaystyle\left|\frac{1}{\delta_{n}}r^{+}_{n}(z)-g(z)\right|$
$\displaystyle\leq\left|\frac{1}{\delta_{n}}r^{+}_{n}(z)-g(z)-C_{D}\parallel
g\parallel_{\alpha}^{2}\delta_{n}^{\frac{\alpha}{2}}\right|+C_{D}\parallel
g\parallel_{\alpha}^{2}\delta_{n}^{\frac{\alpha}{2}}$
$\displaystyle=g(z)-\frac{1}{\delta_{n}}r^{+}_{n}(z)+2C_{D}\parallel
g\parallel_{\alpha}^{2}\delta_{n}^{\frac{\alpha}{2}}.$

Thus, by the triangle inequality, Fubini’s theorem, and the dominated
convergence theorem, (4.14) follows if we show that for all $z\in bD$,

(4.15)
${{\lim_{n\to\infty}\frac{\mathbb{E}[r^{+}_{n}(z)]}{\delta_{n}}=g(z).}}$

Step 3. Expectation of depth. Fix $z\in bD$. Since $\\{r_{n}(z)\geq
s\\}=\\{r^{+}_{n}(z)\geq s\\}$ for $s>0$, we have that,

$\mathbb{E}[r^{+}_{n}(z)]=\int_{0}^{r_{\infty}(z;\boldsymbol{\delta}_{n})}\mathbb{P}(r_{n}(z)\geq
s)\text{d}s.$

Thus, since $r_{\infty}(z;\boldsymbol{\delta}_{n})\geq g(z)\delta_{n}$, we
have that

(4.16)
$0\leq\mathbb{E}[r^{+}_{n}(z)]-\int_{0}^{g(z)\delta_{n}}\mathbb{P}(r_{n}(z)\geq
s)\text{d}s\leq(r_{\infty}(z;\boldsymbol{\delta}_{n})-g(z)\delta_{n}).$

By (3.6), it suffices to prove (4.15) with $\mathbb{E}[r^{+}_{n}(z)]$ replaced
by the integral in (4.16).

Now, recall that $\\{w\in bD:r_{w}(z;\boldsymbol{\delta}(w))\geq
s\\}=G_{s}(z;\boldsymbol{\delta})$, the $(\boldsymbol{\delta},s)$-visibility
region at $z$. Thus,

(4.17)
$\mathbb{P}\left(r_{W^{i}}(z;\boldsymbol{\delta}(W^{i}))<s\right)=\mathbb{P}(W^{i}\notin
G_{s}(z;\boldsymbol{\delta}))=1-\sigma_{f}(G_{s}(z;\boldsymbol{\delta})),$

where recall $\sigma_{f}=f\text{d}\sigma$ is the probability distribution of
the $W^{i}$’s. Therefore,

$\displaystyle\mathbb{P}(r_{n}(z)\geq s)=1-\mathbb{P}(r_{n}(z)<s)$
$\displaystyle=$ $\displaystyle
1-\mathbb{P}\left(\bigcap_{i=1}^{|\eta_{n}|}\\{r_{W^{i}}(z;\boldsymbol{\delta}_{n}(W^{i}))<s\\}\right)$
$\displaystyle=$ $\displaystyle
1-\mathbb{E}\left[\prod_{i=1}^{|\eta_{n}|}\mathbb{P}\Big{(}r_{W^{i}}(z;\boldsymbol{\delta}_{n}(W^{i}))<s\Big{)}\right],$

where, in the last step, we have conditioned on $|\eta_{n}|$ and used the
independence of $|\eta_{n}|$ and $W^{i}$’s. Now, by (4.17), and the fact that
$\mathbb{E}[t^{|\eta_{n}|}]=e^{-n(1-t)}$ for $t\in{\mathbb{R}}$, we have that

(4.18) $\displaystyle\int_{0}^{g(z)\delta_{n}}\mathbb{P}(r_{n}(z)\geq
s)\text{d}s$ $\displaystyle=$
$\displaystyle\int_{0}^{g(z)\delta_{n}}\left(1-\mathbb{E}\Big{[}(1-\sigma_{f}(G_{s}(z;\boldsymbol{\delta}_{n})))^{|\eta_{n}|}\Big{]}\right)\text{d}s$
$\displaystyle=$
$\displaystyle\int_{0}^{g(z)\delta_{n}}\big{(}1-e^{-n\sigma_{f}(G_{s}(z;\boldsymbol{\delta}_{n}))}\big{)}\text{d}s$
$\displaystyle=$ $\displaystyle
g(z)\delta_{n}\int_{0}^{1}\left(1-e^{-n\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))}\right)\text{d}t,$

where, in the final step, we use the change of variable $s\mapsto
tg(z)\delta_{n}$. Now, setting $c_{f}:=\inf_{w\in bD}f(w)$, we have that

$c_{f}\sigma(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))\leq\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))\leq\|f\|_{\infty}\sigma(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n})).$

Now, recall that (3.22) grants constants $C_{D},K_{D}>0$ such that for $n$
large and $t\leq 1-C_{D}\sqrt{\delta_{n}}$,

$|\sigma(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))-h_{d}(t)k_{d}(z)g(z)^{d}\delta_{n}^{d}|\leq
K_{D}\delta_{n}^{d+\alpha/2}.$

Therefore, since $\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$, we obtain
that for any $t<1$,

(4.19) $n\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))\to\infty\
\text{as}\ n\to\infty.$

In other words,
$1-e^{-n\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))}\to 1$ as
$n\to\infty$ for any $t<1$. Thus, by (4.18) and the dominated convergence
theorem, we have that

(4.20)
$\lim_{n\to\infty}\frac{1}{\delta_{n}}\int_{0}^{g(z)\delta_{n}}\mathbb{P}(r_{n}(z)\geq
s)\text{d}s=\lim_{n\to\infty}g(z)\int_{0}^{1}\left(1-e^{-n\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))}\right)\text{d}t=g(z).$

Substituting (4.20) into (4.16) yields (4.15). This completes the proof in the
case of $\eta_{n}$.

Now, assume that $\beta_{n}$ is a binomial point process satisfying (the
analogue of) (2.5). The proof of (2.6) for $\beta_{n}$ is identical to the
above argument until (4.16) as one does not rely on the properties of Poisson
point processes until that stage. For the rest of the proof, it suffices to
show that

(4.21) $\mathbb{P}(r_{n}(z)\geq tg(z)\delta_{n})\rightarrow 1\ \text{as}\
n\rightarrow\infty$

for any $t<1$, where $r_{n}(z)=\max_{1\leq j\leq
n}r_{W^{j}}(z;\boldsymbol{\delta}_{n}(W^{j}))$. Now,

$\displaystyle\mathbb{P}(r_{n}(z)<tg(z)\delta_{n})$
$\displaystyle=\mathbb{P}(\cap_{i=1}^{n}\\{r_{W^{i}}(z;\boldsymbol{\delta}_{n}(W^{i}))<tg(z)\delta_{n}\\})=\big{(}1-\sigma_{f}(G_{tg(z)\delta_{n}}(z;\boldsymbol{\delta}_{n}))\big{)}^{n},$

which converges to $0$ as $n\rightarrow\infty$ for any $t<1$ by (4.19). Thus,
(4.21) holds, and the proof is complete. ∎

###  4.2. Theorem 2.4: $L^{1}$-convergence and variance bounds

###### Proof of Theorem 2.4.

Let $\eta_{n}$ be a Poisson process on $bD$ satisfying (2.7), i.e.,

$\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)=o\left(n^{-\left(1+\frac{2}{d}+\epsilon\right)}\right)\ \text{as}\
n\rightarrow\infty,$

for some $\varepsilon>0$. Recall that
$\delta_{V}(n)=\delta_{V}(P(\eta_{n};\boldsymbol{\delta}_{n}),D)$.

Part 1. $L^{1}$-Convergence. We take expectations on both sides of the
inequality (4.1). The expectations of the second and third terms of the right-
hand side of (4.1) converge to $0$ as $n\rightarrow\infty$ as proved in
Section 4.1; see (4.10) and (4.11). Moreover, by (4.6) and (2.7), we have that

$\mathbb{E}\left|\frac{\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))}{\delta_{n}}\right|\leq\lambda(D)\frac{\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)}{\delta_{n}}=o\left(n^{-\left(1+\frac{1}{d}+\varepsilon\right)}(\log
n)^{-\frac{1}{d}}\right)$

as $n\to\infty$. Thus, (2.8) follows.

Part 2. Variance bounds. The rest of the proof is devoted to establishing
(2.9). This will be done in three steps. First, we approximate the variance of
$\delta_{V}(n)$ by that of $\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))$,
the volume of tubular cuts. This approximation is done by assuming (2.9) for
$\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))$. Next, we derive variance
upper bounds for the volume of tubular cuts. Finally, we derive variance lower
bounds for the volume of tubular cuts. Thus, we prove (2.9) for
$\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))$ and complete the proof.

Step 1. Approximating by tubular cuts. First, we recall the standard variance-
covariance inequality. Let $X,Y$ be random variables such that
$\mathbb{E}[X^{2}],\mathbb{E}[Y^{2}]<\infty$. Then

$\displaystyle|\operatorname{{\mathbb{V}ar}}[X]-\operatorname{{\mathbb{V}ar}}[Y]|$
$\displaystyle=|\operatorname{{\mathbb{V}ar}}[X-Y]+2\operatorname{{\mathbb{C}ov}}(X-Y,Y)|$
(4.22)
$\displaystyle\leq\operatorname{{\mathbb{V}ar}}[X-Y]+2\sqrt{\operatorname{{\mathbb{V}ar}}[X-Y]\operatorname{{\mathbb{V}ar}}[Y]}$

Now, by (4.22) and (4.6),

$\displaystyle\left|\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]-\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]\right|$
$\displaystyle\leq\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]+2\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}$
$\displaystyle\leq\mathbb{E}[|\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))|^{2}]+2\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\delta_{n}))]\mathbb{E}[|\delta_{V}(n)-\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))|^{2}]}$
(4.23)
$\displaystyle\leq\lambda(D)^{2}\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)+2\lambda(D)\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)}.$

We will now complete the proof of (2.9) assuming that there exist
$C_{1},C_{2}>0$, such that

(4.24) $C_{1}\frac{1}{n^{(1+2/d)}(\log
n)^{2(1+1/d)}}\leq\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]\leq
C_{2}\frac{(\log n)^{2(1+1/d)}}{n^{(1+2/d)}}.$

We defer the proof of (4.24) to Step 2 and Step 3 below. Since
$\operatorname{{\mathbb{V}ar}}[X]\leq\mathbb{E}[X^{2}]$, by (4.9), we have
that

$\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]\leq
Cn^{2}\delta_{n}^{4d+2},$

for some $C=C_{g,D}$. Combining this with the lower bound in (4.24) yields
that

$\displaystyle\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]}$
$\displaystyle\leq$ $\displaystyle
Cn^{2}\delta_{n}^{4d+2}\left({\frac{n\delta_{n}^{2(d+1)}}{(\log
n)^{4(d+1)/d}}}\right)^{-1}$ $\displaystyle=$ $\displaystyle C\frac{(\log
n)^{6+\frac{4}{d}}}{n}.$

Thus, we have established

(4.25)
$\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]=o(\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]){\quad\text{as}\
n\rightarrow\infty}.$

Now, (4.25) along with the variance-covariance bound (4.22) implies that

(4.26)
$\lim_{n\to\infty}\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]}=1.$

Then, using assumption (2.7) in (4.23) and combining it with (4.24) and
(4.26), (2.9) follows.

Step 2. Proof of upper bound for
$\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]$
in (4.24). We use the Poincaré inequality i.e., the upper bound in (A.1). Now,

$\displaystyle D_{z}\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))$
$\displaystyle=\lambda(TC(\eta_{n}\cup\\{z\\};\boldsymbol{\delta}_{n})\setminus
TC(\eta_{n};\boldsymbol{\delta}_{n}))=\lambda(TC(z;\boldsymbol{\delta}_{n}(z))\setminus
TC(\eta_{n};\boldsymbol{\delta}_{n})).$

Thus, by (3.13), there is a constant $C>0$ (independent of $z$) such that

(4.27)
$\displaystyle|D_{z}\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))|^{2}\leq\lambda(TC(z;\boldsymbol{\delta}_{n}(z)))^{2}\leq\lambda(C(z;\boldsymbol{\delta}_{n}(z)))^{2}\leq
C\delta_{n}^{2(d+1)}.$

Since $\eta_{n}$ has intensity measure $nf\sigma$, we have from (A.1) that

$\displaystyle\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]$
$\displaystyle\leq Cn\delta_{n}^{2(d+1)}\int_{bD}f(z)\text{d}\sigma(z)\leq
Cn\delta_{n}^{2(d+1)}.$

The upper bound in (4.24) follows since $\delta_{n}=(\log n/n)^{1/d}$.

Step 3. Proof of lower bound for
$\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]$
in (4.24). We shall show that there is a $C>0$ such that

(4.28)
$\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]\geq
C\frac{n\delta_{n}^{2(d+1)}}{(\log n)^{4(d+1)/d}}.$

The lower bound in (4.24) will then follow since $\delta_{n}=(\log
n/n)^{1/d}$.

We use the lower bound in (A.1), for which we must estimate the appropritate
difference operator. Given any $s>0$, we have that

$\displaystyle D_{z}\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))$
$\displaystyle=\lambda(TC(\eta_{n}\cup\\{z\\};\boldsymbol{\delta}_{n})\setminus
TC(\eta_{n};\boldsymbol{\delta}_{n}))$ (4.29)
$\displaystyle\geq\lambda(TC(\eta_{n}\cup\\{z\\};\boldsymbol{\delta}_{n})\setminus
TC(\eta_{n};\boldsymbol{\delta}_{n}))\mathbf{1}[\eta_{n}\cap
S(z,s\boldsymbol{\delta}_{n}(z))=\emptyset].$

By Proposition 3.17, there exist constants $t,B,c>0$ (depending only on $g$
and $D$) such that for any $s<t$, if $\delta_{n}<Bs^{4/\alpha}$, $z\in bD$,
$\varphi=\\{w^{1},...,w^{m}\\}\subset bD$ and $\varphi\cap
S(z;s\boldsymbol{\delta}_{n}(z))=\emptyset$, then

$\displaystyle\lambda\big{(}TC(\phi\cup\\{z\\};\boldsymbol{\delta}_{n})\setminus
TC(\phi;\boldsymbol{\delta}_{n})\big{)}$
$\displaystyle\geq\lambda\left(TC(z;\boldsymbol{\delta}_{n}(z))\setminus\bigcup_{w\in
bD\setminus
S(z;s\boldsymbol{\delta}_{n}(z))}TC(w;\boldsymbol{\delta}_{n}(w))\right)$
$\displaystyle\geq cs^{2(d+1)}\delta_{n}^{d+1}.$

Applying this to (4.2), we obtain that for any $s<t$, if
$\delta_{n}<Bs^{4/\alpha}$, then

(4.30) $D_{z}\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))\geq
cs^{2(d+1)}\delta_{n}^{d+1}\mathbf{1}[\eta_{n}\cap
S(z,s\boldsymbol{\delta}_{n}(z))=\emptyset].$

Now,

$\displaystyle\mathbb{P}(\eta_{n}\cap
S(z,s\boldsymbol{\delta}_{n}(z))=\emptyset)^{2}$
$\displaystyle=\exp\\{-2n\sigma_{f}(S(z,s\boldsymbol{\delta}_{n}(z)))\\}$
$\displaystyle\geq\exp\\{-cn\sigma(S(z,s\boldsymbol{\delta}_{n}(z)))\\}\qquad\mbox{($f$
is bounded above)}$
$\displaystyle\geq\exp\\{-cns^{d}\boldsymbol{\delta}_{n}(z)^{d}\\}\qquad\qquad\quad\mbox{(by
\eqref{eq:cap})}$ $\displaystyle\geq
n^{-cg(z)^{d}s^{d}}\qquad\qquad\qquad\qquad\quad\,(\delta_{n}=(\log
n/n)^{1/d})$ $\displaystyle\geq n^{-\widetilde{c}s^{d}},$

where $\widetilde{c}=c||g||_{\infty}<\infty$. Thus, by (4.30), for some
$c_{1}>0$,

$\displaystyle\int_{bD}\left(\mathbb{E}[D_{z}\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]\right)^{2}\text{d}\sigma_{f}(z)$
$\displaystyle\geq$ $\displaystyle
cs^{4(d+1)}n\delta_{n}^{2(d+1)}\int_{bD}\mathbb{P}(\eta_{n}\cap
S(z,s\boldsymbol{\delta}_{n}(z))=\emptyset)^{2}\text{d}\sigma_{f}(z)$
$\displaystyle\geq$ $\displaystyle
c_{1}s^{4(d+1)}n^{1-cs^{d}}\delta_{n}^{2(d+1)},$

if $\delta_{n}<Bs^{4/\alpha}$ and $n$ is sufficiently large. Thus, invoking
the lower bound in (A.1),

(4.31)
$\operatorname{{\mathbb{V}ar}}[\lambda(TC(\eta_{n};\boldsymbol{\delta}_{n}))]\geq
c_{1}s^{4(d+1)}n^{1-cs^{d}}\delta_{n}^{2(d+1)},$

if $\delta_{n}<Bs^{4/\alpha}$ and $n$ is sufficiently large. Since
$\delta_{n}=(\frac{\log n}{n})^{\frac{1}{d}}$, (4.31) holds when
$s=\left(\frac{4(d+1)}{cd\log(n)}\right)^{\frac{1}{d}}$ and $n$ is
sufficiently large, and yields (4.28). Though we have not used the fact, it
can be verified that this choice of $s$ is the optimal one.

Now, assume that $\beta_{n}$ is a binomial point process satisfying (the
analogue of) (2.7). The proof of $L^{1}$-convergence, i.e., (2.8) for
$\beta_{n}$ follows as in Step 1 above. The upper bound on the variance in
(2.9) can be derived as done in Step 2 above with the following modification.
Instead of the Poincaré inequality, apply the Efron–Stein inequality (A.3) to
$F(\cdot)=\lambda(C(\cdot,\boldsymbol{\delta}_{n}))$, and use the fact that
(4.27) holds for $\lambda(C(\beta_{n},\boldsymbol{\delta}_{n}))$ as well. ∎

###  4.3. Theorem 2.5: central limit theorem

###### Proof of Theorem 2.5.

The strategy of the proof is to reduce (2.11) to a central limit theorem for
$\lambda(C(\varphi_{n};\delta_{n}))$, where $\varphi_{n}$ is either $\eta_{n}$
or $\beta_{n}$. The latter is then handled using second-order Poincaré
inequalities (Theorem A.2 in the case of $\eta_{n}$, and Theorem A.3 in the
case of $\beta_{n}$).

Case I. The source set is $\eta_{n}$. Let $\eta_{n}$ be a Poisson process on
$bD$ satisfying (2.7). Recall that
$\delta_{V}(n)=\delta_{V}(P(\eta_{n};\boldsymbol{\delta}_{n}),D)$, and the
Wasserstein distance is as in (2.10). We will often invoke the following
simple observation.

(4.32)
$d_{W}(X,Y)\leq\sup_{h\in\mathrm{Lip}(1)}\mathbb{E}|h(X)-h(Y)|\leq\mathbb{E}|X-Y|.$

By the triangle inequality, we have that

$d_{W}\Big{(}\frac{\delta_{V}(n)-\mathbb{E}[\delta_{V}(n)]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}},Z\Big{)}\leq\text{I}+\text{II}+\text{III},$

where

I $\displaystyle:=$ $\displaystyle
d_{W}\left(\frac{\delta_{V}(n)-\mathbb{E}[\delta_{V}(n)]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}},\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}\right),$
II $\displaystyle:=$ $\displaystyle
d_{W}\left(\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}},\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}}\right),$
III $\displaystyle:=$ $\displaystyle
d_{W}\left(\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}},Z\right).$

Part 1. Upper bound for I. Using(4.32), we have that

I $\displaystyle\leq$
$\displaystyle\mathbb{E}\left|\frac{\delta_{V}(n)-\mathbb{E}[\delta_{V}(n)]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}-\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}\right|$
$\displaystyle\leq$ $\displaystyle
2\lambda(D)\frac{\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)}{\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}\qquad\qquad\qquad\mbox{(by
\eqref{e:delvvolp} and \eqref{E:upperlowerbounds})}$ $\displaystyle\leq$
$\displaystyle C\frac{n^{-(1+2/d+\epsilon)}}{\sqrt{n^{-(1+2/d)}(\log
n)^{-2(1+1/d)}}}\qquad\qquad\,\,\mbox{(by \eqref{e:covexpvarpoi} and
\eqref{e:varasymptotpoi})}$ $\displaystyle=$ $\displaystyle C\frac{(\log
n)^{(1+1/d)}}{n^{\epsilon+\frac{1}{2}+\frac{1}{d}}}.$

Part 2. Upper bound for II. Denoting
$\dfrac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}}$
by $X_{n}$, we have that

(4.33) II $\displaystyle=$ $\displaystyle
d_{W}\left(\sqrt{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}\,X_{n},X_{n}\right)$
$\displaystyle\leq$
$\displaystyle\left|\sqrt{{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}}-1\right|\mathbb{E}|X_{n}|$
$\displaystyle=$
$\displaystyle\left|{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}-1\right|\frac{1}{\left|\sqrt{{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}}+1\right|}\mathbb{E}|X_{n}|$
$\displaystyle\leq$
$\displaystyle\left|{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}-1\right|,$

where, in the last step, we have used the Cauchy–Schwartz inequality to
observe that,

$\mathbb{E}|X_{n}|=\mathbb{E}\left|\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}}\right|\leq\sqrt{\mathbb{E}\left[\frac{\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))-\mathbb{E}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\sqrt{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}}\right]^{2}}=1.$

Using the variance-covariance bound (4.22) as done in (4.23), we have that

$\displaystyle\left|{\frac{\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}}-1\right|=\frac{|\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]-\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]|}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}$
(4.34)
$\displaystyle\qquad\quad\leq\frac{\lambda(D)^{2}\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)+2\lambda(D)\sqrt{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]\mathbb{P}(P(\eta_{n};\boldsymbol{\delta}_{n})\not\subset
D)}}{\operatorname{{\mathbb{V}ar}}[\delta_{V}(n)]}.$

Combining this with (4.33), and invoking the bounds in (2.7) and (2.9), we
have that

II $\displaystyle\leq$ $\displaystyle
C\left({n^{-(1+2/d+\epsilon)}}{n^{(1+\frac{2}{d})}(\log
n)^{2(1+\frac{1}{d})}}+\sqrt{{n^{-(1+2/d+\epsilon)}}{n^{(1+\frac{2}{d})}(\log
n)^{2(1+\frac{1}{d})}}}\,\right)$ $\displaystyle\leq$ $\displaystyle
C\,\frac{(\log n)^{(1+\frac{1}{d})}}{n^{\frac{\epsilon}{2}}}.$

Part 3. Upper bound for III. We use Theorem A.2, for which must derive bounds
for $T_{i}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))$, $i=1,2,3$, as
defined in Theorem A.2. Note that in the case of $\eta_{n}$, the intensity
measure is $\mu=n\sigma_{f}=nf\sigma$.

To compute $T_{1}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))$ and
$T_{2}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))$, we need to bound the
second-difference operator of $\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))$.
Note that for $z,w\in bD$,

$\displaystyle\Big{|}D^{2}_{z,w}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))\Big{|}$
$\displaystyle=\Big{|}\lambda(C(\eta_{n}\cup\\{z,w\\};\boldsymbol{\delta}_{n}))-\lambda(C(\eta_{n}\cup\\{w\\};\boldsymbol{\delta}_{n}))-\lambda(C(\eta_{n}\cup\\{z\\};\boldsymbol{\delta}_{n}))+\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))\Big{|}$
$\displaystyle\leq\lambda(C(z;\boldsymbol{\delta}_{n}(z))\cap
C(w;\boldsymbol{\delta}_{n}(w)))\qquad\qquad\qquad\,\mbox{(by the inclusion-
exclusion principle)}$ $\displaystyle\leq
C\delta_{n}^{d+1}\mathbf{1}[C(z;\boldsymbol{\delta}_{n}(z))\cap
C(w;\boldsymbol{\delta}_{n}(w))\neq\emptyset]\qquad\mbox{(by \eqref{eq:cut})}$
(4.35) $\displaystyle\leq
C\delta_{n}^{d+1}\mathbf{1}[C(z;\|g\|_{\infty}\delta_{n})\cap
C(w;\|g\|_{\infty}\delta_{n})\neq\emptyset].$

Thus,

$\displaystyle A(z^{1},z^{2},z^{3}):=\,$
$\displaystyle\mathbb{E}\left[\big{(}D^{2}_{z^{1},z^{2}}\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))\big{)}^{2}\big{(}D^{2}_{z^{2},z^{3}}\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))\big{)}^{2}\right]$
$\displaystyle\leq\delta_{n}^{4(d+1)}\mathbf{1}\left[z^{1}\in
N(z^{2};||g||_{\infty}\delta_{n})\right]\mathbf{1}\left[z^{3}\in
N(z^{2};||g||_{\infty}\delta_{n})\right].$

where $N(z;\delta):=\\{w:C(w;\delta)\cap C(z;\delta)\neq\emptyset\\}.$ Also,
from (4.27), we have that

$B(z^{1},z^{2}):=\mathbb{E}[(D^{1}_{z^{1}}\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))^{2}(D^{1}_{z^{2}}\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))^{2}]\leq
C\delta_{n}^{4(d+1)}.$

Therefore, by Proposition 3.16 and the fact that $||g||_{\infty}<\infty$,

$\displaystyle T_{1}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))^{2}$
$\displaystyle=$
$\displaystyle\int_{(bD)^{3}}A(z^{1},z^{2},z^{3})^{1/2}B(z^{1},z^{2})^{1/2}d\mu^{3}(z^{1},z^{2},z^{3})$
$\displaystyle\leq$ $\displaystyle
Cn^{3}\delta_{n}^{4(d+1)}\int_{bD}\sigma\left(N(z;||g||_{\infty}\delta_{n})\right)^{2}\text{d}\sigma_{f}(z)$
$\displaystyle\leq$ $\displaystyle Cn^{3}\delta_{n}^{6d+4}.$

In a similar fashion, we have that

$\displaystyle T_{2}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))^{2}$
$\displaystyle=$
$\displaystyle\int_{(bD)^{3}}A(z^{1},z^{2},z^{3})d\mu^{3}(z^{1},z^{2},z^{3})$
$\displaystyle\leq$ $\displaystyle
Cn^{3}\delta_{n}^{4(d+1)}\int\sigma(N(z;\|g\|_{\infty}\delta_{n}))^{2}\text{d}\sigma_{f}(z)\leq
Cn^{3}\delta_{n}^{6d+4}.$

Finally, an upper bound for
$T_{3}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))$ follows easily from
(4.27):

(4.36) $T_{3}(\lambda(C(\eta_{n};\boldsymbol{\delta}_{n})))\leq
Cn\delta_{n}^{3(d+1)}.$

Now, using that $\delta_{n}=((\log n)/n)^{1/d}$, and noting that the lower
bound in (4.24) holds for
$\operatorname{{\mathbb{V}ar}}[\lambda(C(\eta_{n};\boldsymbol{\delta}_{n}))]$
because of (4.26), we have that

(4.37)