# Hyperbolic width functions and characterizations of bodies of constant width
in the hyperbolic space

Károly J. Böröczky, András Csépai, Ádám Sagmeister  Alfréd Rényi Institute of
Mathematics, Realtanoda u. 13-15, H-1053 Budapest, Hungary
<EMAIL_ADDRESS>Eötvös Loránd University, Institute of Mathematics,
Pázmány Péter sétány 1/c, Budapest, H-1117 Hungary
<EMAIL_ADDRESS>Eötvös Loránd University, Institute of
Mathematics, Pázmány Péter sétány 1/c, Budapest, H-1117 Hungary
<EMAIL_ADDRESS>
###### Abstract.

We discuss basic properties of several different width functions in the
$n$-dimensional hyperbolic space such as continuity, and we also define a new
hyperbolic width as the extension of Leichtweiss’ width function. Then we
prove a characterization theorem of bodies of constant width regarding the
aforementioned notions of hyperbolic width.

###### Key words and phrases:

convex geometry, hyperbolic geometry, bodies of constant width, complete sets,
width function

###### 2010 Mathematics Subject Classification:

Primary:

##  1\. Introduction

Let $H^{n}$ be the $n$-dimensional hyperbolic space for $n\geq 2$. We write
$d(x,y)$ to denote the geodesic distance between $x,y\in H^{n}$ and $[x,y]$ to
denote the geodesic segment between $x$ and $y$ whose length is $d(x,y)$. For
$X\subseteq H^{n}$, we say that $X$ is _convex_ if $[x,y]\subseteq X$ for
$x,y\in X$. In addition, $X$ is a _convex body_ , if $X$ is compact and convex
with non-empty interior. For $z\in H^{n}$ and $r\geq 0$, we write
$B(z,r)=\\{x\in H^{n}\colon d(x,z)\leq r\\}$ to denote the ball of radius $r$
centered at $z$.

Bodies of constant width play an important role in the well-researched field
of convex geometric inequalities in the Euclidean space or in Minkowski spaces
(see Martini, Swanepoel[11]). Some more recent studies investigate bodies of
constant width on the sphere (see Lassak, Musielak[9]). However, when it comes
to the hyperbolic space, there are many different attempts to define the width
function of a convex body (see Santaló[12], Fillmore[6], Leichtweiss[10],
Jerónimo-Castro–Jimenez-Lopez[8], G. Horváth[7]). We will also introduce a new
concept of hyperbolic width, based on Leichtweiss’ definition. Sometimes we
cannot even compare these functions, and even when we can, they often times
take different values. The goal of this paper is to show, that this difference
disappears within the class of bodies of constant width.

Organisation of the paper  In section 2 we introduce the models of the
hyperbolic space we will use throughout the paper. In section 3 we describe
basic properties of compact subsets in $H^{n}$. In section 4 we define the
width functions $w_{S}$, $w_{F}$, $w_{L}$, $w_{JCJL}$, $w_{GH}$ of Santaló,
Fillmore, Leichtweiss, Jerónimo-Castro–Jimenez-Lopez and G. Horváth and
introduce the new width function $w$. Section 5 is about the continuity of
these width functions, in particular we prove:

######  Theorem 1.1.

The width functions $w_{S}(K,(z,H_{z})$, $w_{F}(K,i)$, $w_{L}(K,(p,H_{p}))$,
$w_{JCJL}(K,(z,\ell_{z},z^{\prime}))$, $w_{GH}(K,i)$ and $w(K,H)$ are
continuous in all of their parameters. The minimal widths $w_{S}(K)$,
$w_{F}(K)$, $W_{L}(K,p)$, $w_{JCJL}(K)$, $w_{GH}(K)$ and $w(K)$ of convex
bodies $K$ are also continuous in $K$.

In section 6 we verify that the diameter coincides with the maximal width
using any of the width notions defined in the paper.

Section 7 deals with complete bodies and bodies of constant width. It contains
our main result:

######  Theorem 1.2.

Let $K\subset H^{n}$ be a convex body of diameter $D>0$. The following are
equivalent:

(i):

$K$ is complete;

(ii):

$K=\bigcap_{x\in K}B\left(x,D\right)$;

(iii):

$K$ is of constant width $D$;

(iv):

$w_{S}\left(K,\left(z,H_{z}\right)\right)=D$ for all $z\in\partial K$ and
$H_{z}$ supporting hyperplanes to $K$ at $z$;

(v):

$w_{F}\left(K,i\right)=D$ for all ideal points $i$;

(vi):

$w_{L}\left(K,\left(p,H_{p}\right)\right)=D$ for all hyperplanes through a
fixed inner point $p\in\text{\rm int }K$;

(vii):

$w\left(K,H\right)=D$ for all hyperplanes $H$ intersecting $K$.

Finally, in section 8 we consider the circumscribed and inscribed balls of
convex bodies and deduce results on the circumradii and inradii of complete
bodies. In particular, we prove a hyperbolic version of Scott’s theorem:

######  Theorem 1.3.

Let $K\subset H^{n}$ be a convex body of diameter $D$ and circumscribed ball
$B$. Then, there exists a completion $\widetilde{K}$ of $K$ such that $B$ is
also the circumscribed ball of $\widetilde{K}$.

##  2\. Models of the hyperbolic space

###  2.1. The hyperboloid model

We assume that $H^{n}$ is embedded into ${\mathbb{R}}^{n+1}$ using the
hyperboloid model. We write $\langle\cdot,\cdot\rangle$ to denote the standard
scalar product in ${\mathbb{R}}^{n+1}$, and write
$z^{\bot}=\\{x\in{\mathbb{R}}^{n+1}\colon\langle x,z\rangle=0\\}$ for a
$z\in{\mathbb{R}}^{n+1}\setminus\left\\{o\right\\}$. Fix an $e\in S^{n}$, then
we have

$H^{n}=\left\\{x+te\colon x\in e^{\bot}\mbox{ and }t\geq 1\mbox{ and
}t^{2}-\langle x,x\rangle=1\right\\}.$

We also consider the following symmetric bilinear form $\mathcal{B}$ on
${\mathbb{R}}^{n+1}$: If $x=x_{0}+te\in{\mathbb{R}}^{n+1}$ and
$y=y_{0}+se\in{\mathbb{R}}^{n+1}$ for $x_{0},y_{0}\in e^{\bot}$ and
$t,s\in{\mathbb{R}}$, then

$\mathcal{B}(x,y)=ts-\langle x_{0},y_{0}\rangle.$

In particular,

(1) $\mathcal{B}(x,x)=1\mbox{ for $x\in H^{n}$}.$

For $z\in H^{n}$, we define the tangent space $T_{z}$ as

$T_{z}=\left\\{x\in{\mathbb{R}}^{n+1}\colon\mathcal{B}(x,z)=0\right\\}.$

We observe that $T_{z}$ is an $n$-dimensional real vector space equipped with
the scalar product $-\mathcal{B}(\cdot,\cdot)$.

For $z\in H^{n}$ and unit vector $u\in T_{z}$, the geodesic line $\ell$
passing through $z$ and determined by $u$ consists of the points

$p_{t}=z\,{\rm cosh}\,t+u\,{\rm sinh}\,t$

for $t\in{\mathbb{R}}$. Here the map $t\mapsto p_{t}$ is bijective onto $\ell$
and satisfies $d(z,p_{t})=|t|$ for $t\in{\mathbb{R}}$. If $t>0$, then we say
that $u$ points towards $p_{t}$ along the geodesic segment

$[z,p_{t}]=\\{p_{s}\colon 0\leq s\leq t\\}$

of length $t$.

A hyperplane $H$ in $H^{n}$ passing through the point $z\in H^{n}$ and having
exterior unit normal $u\in T_{z}$, and the corresponding half-spaces $H^{+}$
and $H^{+}$ are defined as follows: $H^{-}=H^{n}\setminus{\rm int}H^{+}$,
$H=\\{x\in H^{n}\colon\mathcal{B}(x,u)=0\\}$ and $H^{+}=\\{x\in
H^{n}\colon-\mathcal{B}(x,u)\geq 0\\}$.

We also consider horoballs and horospheres “centered” at ideal points. An
_ideal point_ of $H^{n}$ is represented by the linear hull of a
$z\in{\mathbb{R}}^{n+1}$ with $\mathcal{B}(z,z)=0$ and $\mathcal{B}(z,e)>0$.
Any hyperbolic line $\ell=\Pi\cap H^{n}$ for a linear two-plane $\Pi$ with
$\Pi\cap H^{n}\neq\emptyset$ contains exactly two ideal points. If $p\in\ell$
and $v\in T_{p}\cap\Pi$ is one of the two tangent vectors to $\ell$ at $p$
with $\mathcal{B}(v,v)=-1$, then the two ideal points are represented by the
spans of $u-v$ (“point of $\ell$ at infinity” in the direction of $v$) and
$u+v$ (“point of $\ell$ at infinity” in the direction of $-v$). We also
observe that hyperbolic lines containing either ideal point of $\ell$ are the
lines parallel to $\ell$.

Let $z\in{\mathbb{R}}^{n+1}$ with $\mathcal{B}(z,z)=0$ and
$\mathcal{B}(z,e)>0$, and let $i$ be the ideal point represented by ${\rm
lin}\,z$. For any $p\in H^{n}$, there exists a unique line $\ell$ passing
through $p$ and $i$, and a unit tangent vector $v\in T_{p}$ to $\ell$ is
$u-\mathcal{B}(p,z)^{-1}z$.

For $s>0$, a _horoball_ at $i$ is

$A=\\{x\in H^{n}\colon\mathcal{B}(z,x)\geq s\\},$

and the corresponding _horosphere_ is

$\partial A=\\{x\in H^{n}\colon\mathcal{B}(z,x)=s\\}.$

We observe that for any $p\in\partial A$, the line passing through $p$ and $i$
is orthogonal to the horosphere $\partial A$. If $n=2$, then a horoball is
also called a _horodisk_ and a horosphere is called a _horocycle_ as well.

###  2.2. The Poincaré ball model

We can obtain the Poincaré ball model by a radial projection of the
hyperboloid through the apex of the hyperboloid $e\in{\mathbb{R}}^{n+1}$ onto
$e^{\perp}$. In the Poincaré ball model, the hyperbolic space $H^{n}$ is
identified with the interior of the unit Euclidean ball $B^{n}$ in
${\mathbb{R}}^{n}$, and the set of ideal points are just $\partial B^{n}$. A
hyperbolic line in the Poincaré ball model is the intersection of ${\rm
int}B^{n}$ and a Euclidean circle that is orthogonal to $\partial B^{n}$ at
the two intersection points. A hyperplane is the intersection of a Euclidean
$(n-1)$-sphere with ${\rm int}B^{n}$ orthogonal to $\partial B^{n}$ at the
$(n-2)$-sphere of intersection points. A horosphere at an ideal point
$i\in\partial B^{n}$ is of the form $\partial G\setminus\left\\{i\right\\}$
for a Euclidean $n$-ball $G\subset B^{n}$ of radius less than one and touching
$\partial B^{n}$ in $i$. In addition, hyperbolic $n$-balls, in the Poincaré
model coincide with Euclidean $n$-balls contained in ${\rm int}\,B^{n}$.

###  2.3. The Cayley–Beltrami–Klein model

The Cayley–Beltrami–Klein model can be derived from the hyperboloid model by a
radial projection onto $e^{\perp}-e$ through the origin
$o\in{\mathbb{R}}^{n+1}$. Here, the points of the hyperboloid are the interior
points of the Euclidean unit ball centered at $-e$, while ideal points are
again the points of this unit ball. Sometimes it is easier to investigate
convexity, as hyperplanes and lines of $H^{n}$ are intersections of the ball
with Euclidean hyperplanes and lines. In particular, hyperbolic convexity
coincides with Euclidean convexity in the Cayley–Beltrami–Klein ball.

##  3\. Compact subsets in the hyperbolic space

We impose a metric on compact subsets. For a compact set $C\subset H^{n}$ and
$z\in H^{n}$, we set $d(z,C)=\min_{x\in C}d(z,x)$. For any non-empty compact
set $C_{1},C_{2}\subset H^{n}$, we define their _Hausdorff distance_

$\delta(C_{1},C_{2})=\max\left\\{\max_{x\in C_{2}}d(x,C_{1}),\max_{y\in
C_{1}}d(y,C_{2})\right\\}.$

The Hausdorff distance is a metric on the space of compact subsets in $H^{n}$.
We say that a sequence $\left\\{C_{m}\right\\}_{m\in{\mathbb{N}}}$ of compact
subsets of $H^{n}$ is _bounded_ if there is a ball containing every $C_{m}$.
For compact sets $C_{m},C\subset H^{n}$, we write $C_{m}\to C$ to denote if
the sequence $\left\\{C_{m}\right\\}_{m\in{\mathbb{N}}}$ tends to $C$ in terms
of the Hausdorff distance.

The following Lemma 3.1 is a simple consequence of these properties of the
Poincaré disk model.

######  Lemma 3.1.

Any horoball $A$ in $H^{n}$ is closed and convex, and if $A$ and a ball
$B(z,r)$, $r>0$, have a common exterior normal $u\in T_{y}$ for some
$y\in\partial A\cap\partial B(z,r)$, then $B(z,r)\subset A$ with $B(z,r)\cap
A=\\{y\\}$.

Lemma 3.1 directly yields the following.

######  Corollary 3.2.

If $A$ and $C$ are horoballs at an ideal point $i$ of $H^{n}$ with $C\subset
A$, then $d(x,y)\geq w\left(A\setminus{\rm int}C\right)$ for any $x\in\partial
A$ and $y\in\partial C$ with equality if and only if the line $\ell$ passing
through $x$ and $y$ has $i$ as an ideal point (and hence $\ell$ is orthogonal
to $\partial A$ and $\partial C$).

The following is well-known (see Böröczky, Sagmeister [1]).

######  Lemma 3.3.

For compact sets $C_{m},C\subset H^{n}$, we have $C_{m}\to C$ if and only if

(i):

assuming $x_{m}\in C_{m}$, the sequence $\\{x_{m}\\}$ is bounded and any
accumulation point of $\\{x_{m}\\}$ lies in $C$;

(ii):

for any $y\in C$, there exist $x_{m}\in C_{m}$ for each $m$ such that
$\lim_{m\to\infty}x_{m}=y$.

The space of compact subsets of $H^{n}$ is locally compact according to the
Blaschke Selection Theorem (see R. Schneider [13]).

######  Theorem 3.4 (Blaschke).

Any bounded sequence of compact subsets of $H^{n}$ has a convergent
subsequence.

For convergent sequences of compact subsets of $H^{n}$, we have the following
(see Böröczky, Sagmeister [1]).

######  Lemma 3.5.

Let the sequence $\\{C_{m}\\}$ of compact subsets of $H^{n}$ tend to $C$.
Then,

${\rm diam}\,C=\lim_{m\to\infty}{\rm diam}\,C_{m}.$

Recall that for any compact set $X\subset H^{n}$ and $\varrho\geq 0$, the
parallel domain is

$X^{(\varrho)}=\\{z\in H^{n}\colon\exists x\in X\mbox{ \ with \
}d(x,z)\leq\varrho\\}=\bigcup\\{B(x,\varrho)\colon x\in X\\}.$

The triangle inequality and considering $x,y\in X$ with $d(x,y)={\rm diam}\,X$
show that

(2) ${\rm diam}\,X^{(\varrho)}=2\varrho+{\rm diam}\,X.$

The following is well-known (see Böröczky, Sagmeister[3]):

######  Lemma 3.6.

If $\varrho\geq 0$, and the sequence $\\{C_{m}\\}$ of compact subsets of
$H^{n}$ tends to $C$, then

(i):

$\left\\{C_{m}^{(\varrho)}\right\\}$ tends to $C^{(\varrho)}$;

(ii):

${\rm diam}\,C^{(\varrho)}=\lim_{m\to\infty}{\rm diam}\,C_{m}^{(\varrho)}$.

The following lemma shows that it is true, that the parallel domain of a
convex body is also a convex body in the hyperbolic space.

######  Lemma 3.7.

Let $K\subset H^{n}$ be a convex body of diameter $D>0$ and $\varepsilon>0$.
Then the parallel set $K^{\left(\varepsilon\right)}$ is also a convex body.

###### Proof.

It is clear that to verify that $K^{\left(\varepsilon\right)}$ is a convex
body, we only need to prove its convexity. We use the Cayley–Beltrami–Klein
model to verify that. In this model, hyperbolic lines are Euclidean lines,
therefore it is enough to check Euclidean convexity in this model. Let $x,y\in
K^{\left(\varepsilon\right)}$ and $\lambda\in\left[0,1\right]$. We want to see
that $\lambda x+\left(1-\lambda\right)y\in K^{\left(\varepsilon\right)}$. By
the definition of the parallel set, there are points $x_{0},y_{0}\in K$, such
that $d\left(x,x_{0}\right)\leq\varepsilon$ and
$d\left(y,y_{0}\right)\leq\varepsilon$. Let $\ell$ be the line connecting $x$
and $y$. It is well-known, that hypercycles are convex in the
Cayley–Beltrami–Klein disk, and $x_{0},y_{0}$ are points in the hypercycle
$\ell^{\left(\varepsilon\right)}$, so the geodesic segment
$\left[x_{0},y_{0}\right]\subset\ell^{\left(\varepsilon\right)}$. Now let
$z\in\left[x_{0},y_{0}\right]$ an arbitrary point, and let $x_{1}$ and $y_{1}$
be the intersection points of the line through $x_{0}$ and $y_{0}$, and the
orthogonal hyperplane of $\ell$ through the points $x$ and $y$, respectively.
We may assume that $\ell$ is a diameter of the disk, so the orthogonal
hyperplane is orthogonal in the Euclidean setup as well. If
$z\in\left[x_{0},y_{0}\right]\cap\left[x_{1},y_{1}\right]\subset\ell^{\left(\varepsilon\right)}$,
then it is clear, that $d\left(z,z^{\prime}\right)\leq\varepsilon$, where $z$
is the intersection point of $\ell$ and the orthogonal hyperplane to $\ell$
through $z$. The only questionable case is when $z\in\left[x_{0},x_{1}\right]$
or $z\in\left[y_{0},y_{1}\right]$. We may assume that the first is true. But
then $z$ is on the side $\left[x_{0},x_{1}\right]$ of the hyperbolic triangle
$\left[x,x_{0},x_{1}\right]$, and both of the sides $\left[x,x_{0}\right]$ and
$\left[x,x_{1}\right]$ are at most $\varepsilon$ long, and hence
$d\left(x,z\right)\leq\varepsilon$ in this case.

∎

##  4\. Hyperbolic width functions

###  4.1. Santaló’s width function

The following width function was introduced by Santaló in $H^{2}$, but has a
natural extension on higher dimensions. Let $K\subset H^{n}$ a convex body.
Let $H_{z}$ be a supporting hyperplane to $K$ at a boundary point
$z\in\partial K$ and $\ell$ be the unique hyperbolic line orthogonal to
$H_{z}$ through $z$. Then there is a unique supporting hyperplane
$H^{\prime}_{z}$ to $K$, which is orthogonal to $\ell$ and is disjoint from
$H_{z}$. If $\left\\{z^{\prime}\right\\}=H^{\prime}_{z}\cap\ell$, we call
$d\left(z,z^{\prime}\right)$ the _Santaló width_ of $K$ at
$\left(z,H_{z}\right)$, and we use the notation

$w_{S}\left(K,\left(z,H_{z}\right)\right).$

The following example shows that although Santaló’s definition is a very
natural extension of the Euclidean width function, it is not monotonous,
considering the minimal width. Let

$w_{S}\left(K\right)=\min\left\\{w_{S}\left(K,\left(z,H_{z}\right)\right)\colon
z\in\partial K\text{ and }H_{z}\text{ is a supporting hyperplane of }K\text{
at }z\right\\}.$

######  Proposition 4.1.

There is a convex body $K\subset H^{2}$ containing the disk
$B=B\left(z,r\right)\subset H^{2}$, such that
$w_{S}\left(K\right)<w_{S}\left(B\right)$.

###### Proof.

There are tangent ultraparallel lines $\ell_{1}$ and $\ell_{2}$ to the disk
$B$, such that the unique perpendicular line $\ell$ to $\ell_{1}$ and
$\ell_{2}$ both is exterior to $B$. Let $y_{1}$ be the intersection point of
$\ell$ and $\ell_{1}$, while $y_{2}$ denotes the intersection point of $\ell$
and $\ell_{2}$. If $\ell_{1}$ touches $B$ at $z_{1}$ and $\ell_{2}$ touches
$B$ at $z_{2}$, then

$w_{S}\left(B\right)=2r=d\left(z,z_{1}\right)+d\left(z,z_{2}\right)>d\left(z_{1},z_{2}\right)>d\left(y_{1},y_{2}\right)=w_{S}\left(\widetilde{K},\left(y_{1},\ell_{1}\right)\right)\geq
w_{S}\left(\widetilde{K}\right),$

where
$\widetilde{K}=\operatorname{\mathrm{conv}}B\cup\left[y_{1},y_{2}\right]$. ∎

Figure 1. The minimal Santaló width is not monotonic

Remark Similarly we can construct a counterexample in $H^{n}$ for any
dimension $n\geq 2$.

###  4.2. Fillmore’s width function

We call a set _h-convex_ , if it is the intersection of some horoballs. A set
in $H^{n}$ is a _h-convex body_ if it is h-convex, compact, and of non-empty
interior. As horoballs are convex, every h-convex body is also a convex body,
but the reverse is not true in general, polytopes for example are convex but
not h-convex.

If $r>s$, then the horoball $C=\\{x\in H^{n}\colon\mathcal{B}(z,x)\geq
r\\}\subset A$ satisfies that the distance of the parallel horospheres
$\partial A$ and $\partial C$ is $d>0$ where $d=\ln\left(\frac{r}{s}\right)$.
In particular, any hyperbolic line $\ell$ containing $i$ satisfies that the
distance between $\ell\cap\partial A$ and $\ell\cap\partial C$ is $d$, and we
set $d=w_{\mathrm{hor}}\left(\partial A,\partial C\right)$ be the width of
closed region bounded by the horospheres $\partial A$ and $\partial C$. This
concept of horospherical width was first introduced in $H^{2}$ by Fillmore
[6], but it has a natural extension to $H^{n}$.

Let $K\subset H^{n}$ be a convex body, and $A\subset H^{n}$ a horoball. Then
$\partial A$ is called a supporting horosphere to $K$, if $\partial
A\cap\partial K\neq\emptyset$ and either $K\subset A$ or $K\subset
H^{n}\setminus\text{\rm int }A$. For an ideal point $i$ and a convex body
$K\subset H^{n}$, there are exactly two parallel supporting horospheres
orthogonal to $i$. If the horoballs determined by these supporting horospheres
are $A$ and $C$, then the _Fillmore-width_ of $K$ at $i$ is

$w_{F}\left(K,i\right)=w_{\mathrm{hor}}\left(\partial A,\partial C\right).$

Remark In contrast with the Santaló width, for the Fillmore-width it is
trivially true, that for an ideal point $i$ and two convex bodies $K\subseteq
L$, $w_{F}\left(K,i\right)\leq w_{F}\left(L,i\right)$. However, if
$w_{F}\left(K,i\right)=w_{F}\left(L,i\right)$ for every ideal point $i$ and
convex bodies $K\subseteq L$, it is not necessarily true, that $K=L$, only
that the h-convex hull of $K$ and $L$ are the same. Note that in general it is
not true, that for every boundary point there is a supporting horospherical
strip (e.g. polytopes are counterexamples), only for h-convex bodies.

Let

$w_{F}\left(K\right)=\min\left\\{w_{F}\left(K,i\right)\colon i\text{ is an
ideal point}\right\\}.$

The following is easy to see (for example let $C_{m}$ be an isosceles triangle
with $s$ as its base and base altitude of length $\frac{1}{m}$, or
$C_{m}=s^{\left(\frac{1}{m}\right)}$).

######  Proposition 4.2.

Let $s=\left[x,y\right]$ be a hyperbolic line segment, and
$\left\\{C_{m}\right\\}_{m\in{\mathbb{N}}}$ a sequence of convex bodies
tending to $s$ with respect to the Hausdorff metric. Then,

$\lim_{m\to\infty}w_{F}\left(C_{m}\right)>0.$

###  4.3. Leichtweiss’ width function

In $H^{n}$, for a hyperplane $H$, the _hyperball_ of radius $\varepsilon\geq
0$ is the parallel domain $H^{\left(\varepsilon\right)}$. The boundary
components of a hyperball are called _hyperspheres_ (or _hypercycles_ in the
2-dimensional case). A hypersphere $\partial\mathcal{H}$ of the hyperball $H$
has two connected components, $H_{1}$ and $H_{2}$ (except the degenerate case
where $\varepsilon=0$ and $H=H_{1}=H_{2}$). Both $H_{1}$ and $H_{2}$ divide
the hyperbolic space $H^{n}$ into two closed connected components, $H_{k}^{+}$
and $H_{k}^{-}$ for $k\in\left\\{1,2\right\\}$, such that $H_{k}^{+}\cap
H_{k}^{-}=H_{k}$, $H_{1}^{+}\cap H_{2}^{+}$ is convex, and $H_{k}^{-}\subseteq
H_{3-k}^{+}$. For a convex body $K\subset H^{n}$ and a hyperplane $H$ such
that $H\cap K\neq\emptyset$, we say that the closed convex set $\mathcal{A}$
is the _supporting hyperspherical domain_ with respect to $H$, if
$K\subset\mathcal{A}$, and the connected components of $\partial\mathcal{A}$
are $H_{1}$ and $H^{\prime}_{2}$, where $H_{1}$ and $H^{\prime}_{2}$ are
components of the hyperspheres $\partial H^{\left(\varepsilon\right)}$ and
$\partial H^{\left(\varepsilon^{\prime}\right)}$, such that $H_{1}\cap
K\neq\emptyset$ and $H^{\prime}_{2}\cap K\neq\emptyset$. Then the width of the
supporting hyperspherical domain $\mathcal{A}$ with respect to the hyperplane
$H$ is $w_{hyp}\left(K,H\right)=\varepsilon+\varepsilon^{\prime}$. The
following notion of hyperspherical width was first used by Leichtweiss[10] in
$H^{2}$.

For a convex body $K\subset H^{n}$ and a fixed interior point $p\in\text{\rm
int }K$, we define the _Leichtweiss width_ of $K$ with respect to a hyperplane
$H$ containing $p$ as

$w_{L}\left(K,\left(p,H\right)\right)=w_{hyp}\left(K,H\right)$

and we put

$w_{L}\left(K,p\right)=\min\left\\{w_{L}\left(K,\left(p,H\right)\right)\colon
H\text{ is a hyperplane containing }p\right\\}.$

Remark We can observe, that once again, there is no supporting hyperspherical
domain through all boundary points necessarily (for example polytopes). It is
trivial, that for convex bodies $K_{1}\subseteq K_{2}$ and $p\in\text{\rm int
}K_{1}$, $w_{L}\left(K_{1},\left(p,H\right)\right)\leq
w_{L}\left(K_{2},\left(p,H\right)\right)$ for any hyperplane $H$ through $p$,
so $w_{L}\left(K_{1},p\right)\leq w_{L}\left(K_{2},p\right)$. Note that
$w_{L}\left(K,p\right)$ depends on the choice of $p$, for instance, if $K$ is
an equilateral triangle in $H^{2}$ and $\ell$ is one of its altitudes, then
$w_{L}\left(K,p\right)$ gets greater as $p\in\ell$ gets closer to the vertex,
and gets smaller (approximating the altitude of the triangle) as $p$ gets
closer to the midpoint of the opposite side.

###  4.4. Planar alternatives

There are two more recent approaches to define hyperbolic width on the plane
$H^{2}$. The first one is by Jerónimo-Castro and Jimenez-Lopez[8]. If
$z\in\partial K$ is a boundary point and $\ell_{z}$ is a supporting line
through $z$, we call a $z^{\prime}\in\partial K$ an _opposite point_ to $z$
with respect to $\ell_{z}$, if there is a supporting line $\ell_{z^{\prime}}$
such that $\ell_{z}$ and $\ell_{z^{\prime}}$ have the same alternating angles
with the geodesic segment $\left[z,z^{\prime}\right]$. By definition, the
_Jerónimo-Castro–Jimenez-Lopez_ width of $K$ at
$\left(z,\ell_{z},z^{\prime}\right)$ is

$w_{JCJL}\left(K,\left(z,\ell_{z},z^{\prime}\right)\right)=d\left(\ell_{z},\ell_{z^{\prime}}\right).$

Note that for a convex body $K$ and a supporting line $\ell_{z}$ through
$z\in\partial K$ the opposite point $z^{\prime}$ is not unique in general, but
if $K$ is h-convex, then it is unique (see [8]).

Another more recent planar width function is defined by G. Horváth[7]. For a
convex body $K\subset H^{2}$ and an ideal point $i$, we define the _G. Horváth
width_ as

$w_{GH}\left(K,i\right)=\sup\left\\{w_{\mathrm{hyp}}\left(\ell\right)\colon\ell\cap\text{\rm
int }K\neq\emptyset\text{ and }i\text{ is an ideal point of }\ell\right\\}.$

We put

$w_{JCJL}(K)=\min\\{w_{JCJL}(K,(z,\ell_{z},z^{\prime}))\colon
z,z^{\prime}\in\partial K\text{ are opposite with respect to }\ell_{z}\\}$

and

$w_{GH}(K)=\min\\{w_{GH}(K,i)\colon i\text{ is an ideal point}\\}.$

###  4.5. A new concept of width

We will extend the width function $w_{hyp}$ introduced by Leichtweiss. Let
$K\subset H^{n}$ be a convex body and let $H$ be a hyperplane intersecting
$K$. We extend the notion of supporting hyperspherical domain $\mathcal{A}$
the following way. If $H$ is a supporting hyperplane, then there is a positive
$\varepsilon$, such that $K\subset H^{\varepsilon}$, and $\partial K$
intersects one of the connected components of $\partial H^{\varepsilon}$ (say
$H_{1}$). Then, the supporting hyperspherical domain $\mathcal{A}$ will be
bounded by $H$ and $H_{1}$, and the corresponding width is
$w_{hyp}\left(H\right)=\varepsilon$ in this case. Now let $H$ be an arbitrary
hyperplane intersecting $K$. Then the _width_ of $K$ with respect to the
hyperplane $H$ is $w\left(K,H\right)=w_{hyp}\left(K,H\right)$.

We define the minimal width of $K$ as

$w(K)=\min\\{w(K,H)\colon H\text{ is a hyperplane intersecting }K\\}$

Remark  It is clear, that for $n=2$ we have

$w_{GH}(K,i)=\max\\{w(K,\ell)\colon i\text{ is an ideal point of }\ell\\}$

##  5\. Continuity of width functions

It is natural to ask whether the above defined notions of width are
continuous. To answer this we first have to define what we mean by continuity
for these objects. In the following we will introduce the necessary
definitions and then the rest of this section concerns the proof of Theorem
1.1, i.e. that the width functions are indeed continuous in every case. We
will work in the Poincaré disk model.

We say that a sequence of ideal points $i_{m}$ converges to an ideal point $i$
if it does so in the usual topology of the sphere $\partial B^{n}$. This
convergence is well-defined since the identification of $H^{n}$ with
$\mathrm{int}B^{n}$ is a homeomorphism and so if we choose another Poincaré
disk model for $H^{n}$, the convergent ideal point sequences remain the same.

Let $H_{m},H$ be hyperplanes with ideal point sets $I_{m},I\subset\partial
B^{n}$ respectively and let $c_{m},c$ and $r_{m},r$ be the centres and radii
of $I_{m},I$ in the sphere $\partial B^{n}$ respectively. Note that the
centre-radius pair $(-c,\pi-r)$ yields the same hyperplane as the pair $(c,r)$
(where we think of $B^{n}$ as the unit ball in ${\mathbb{R}}^{n}$ and $-c$
denotes the reflection of $c$ to the origin). Then the sequence $H_{m}$
converges to $H$ if $\\{(c_{m},r_{m}),(-c_{m},\pi-
r_{m})\\}\to\\{(c,r),(-c,\pi-r)\\}$. Observe that although the centres and
radii of $I_{m}$ and $I$ in $\partial B^{n}$ can change when changing the
model for $H^{n}$, the notion of convergence is again independent of the model
we choose.

If $\ell_{m}$ and $\ell$ are lines with ideal points $i_{m},j_{m}$ and $i,j$
respectively, then the sequence $\ell_{m}$ converges to $\ell$ if
$\\{i_{m},j_{m}\\}\to\\{i,j\\}$.

Remark  With the above definitions the space of ideal points is $\partial
B^{n}$ which is compact. The space of hyperplanes is

$\mathcal{H}:={\raisebox{1.99997pt}{$(\partial
B^{n}\times(0,\pi))$}\left/\raisebox{-1.99997pt}{$(c,r)\sim(-c,\pi-r)$}\right.}$

and the space of lines is

$\mathcal{L}:={\raisebox{1.99997pt}{$\\{(i,j)\in\partial B^{n}\times\partial
B^{n}\colon i\neq j\\}$}\left/\raisebox{-1.99997pt}{$(i,j)\sim(j,i)$}\right.}$

and both are locally compact.

We will use the following basic topological lemma.

######  Lemma 5.1.

Let $X$ be a locally compact and $Y$ a compact topological space, let $f\colon
X\times Y\to{\mathbb{R}}$ be a continuous function and for each $x\in X$ fix a
non-empty closed subset $Z_{x}\subset Y$. Then the function

$F\colon X\to{\mathbb{R}};~{}x\mapsto\max_{y\in Z_{x}}f(x,y)$

is also continuous.

We will usually use this lemma with $Z_{x}=Y$ for all $x$.

###### Proof.

Let $x_{m}\to x$ be a convergent sequence in $X$ and fix an $\varepsilon>0$;
we want to prove that if $m$ is sufficiently large, then

$|\max_{y\in Z_{x}}f(x,y)-\max_{y\in Z_{x_{m}}}f(x_{m},y)|<\varepsilon.$

Since $X$ is locally compact, we may assume that $x$ and $x_{m}$ are all in a
compact subspace $C\subset X$. Now $C\times Y$ is compact, hence there are
open subsets $U\subset C$ and $V_{1},\ldots,V_{n}\subset Y$ such that $x\in U$
and for all $i$ if $a,b\in U\times V_{i}$, then $|f(a)-f(b)|<\varepsilon$. Let
$y_{m},y\in Y$ be points such that

$f(x_{m},y_{m})=\max_{Z_{x_{m}}}f(x_{m},\cdot)\quad\text{and}\quad
f(x,y)=\max_{Z_{x}}f(x,\cdot).$

If $m$ is sufficiently large, then $x_{m}\in U$ which implies
$|f(x,y_{m})-f(x_{m},y_{m})|<\varepsilon$ and
$|f(x_{m},y)-f(x,y)|<\varepsilon$. Now if $f(x,y)\geq f(x_{m},y_{m})$, then we
get

$f(x_{m},y_{m})-\varepsilon\leq f(x,y)-\varepsilon<f(x_{m},y)\leq
f(x_{m},y_{m})$

thus $|f(x,y)-f(x_{m},y_{m})|<\varepsilon$; similarly if $f(x_{m},y_{m})\geq
f(x,y)$, we get

$f(x,y)-\varepsilon\leq f(x_{m},y_{m})-\varepsilon<f(x,y_{m})\leq f(x,y)$

so again $|f(x,y)-f(x_{m},y_{m})|<\varepsilon$. This is what we wanted to
prove. ∎

######  Lemma 5.2.

Let $K\subset H^{n}$ be a convex body, $\ell_{m}\to\ell$ a convergent sequence
of lines with ideal points $i_{m}\to i$, $j_{m}\to j$ and let
$H_{m},H^{\prime}_{m}$ (resp. $H,H^{\prime}$) be the supporting hyperplanes of
$K$ orthogonal to $\ell_{m}$ (resp. $\ell$) such that
$H_{m}\cap\ell_{m}\in[i_{m},H^{\prime}_{m}\cap\ell_{m}]$ and
$H\cap\ell\in[i,H^{\prime}\cap\ell]$. Then with the notations
$\\{a_{m}\\}=H_{m}\cap\ell_{m}$,
$\\{a^{\prime}_{m}\\}=H^{\prime}_{m}\cap\ell_{m}$ and $\\{a\\}=H\cap\ell$,
$\\{a^{\prime}\\}=H^{\prime}\cap\ell$ we have
$d_{H^{n}}(a_{m},a^{\prime}_{m})\to d_{H^{n}}(a,a^{\prime})$.

###### Proof.

Let $I\subset\partial B^{n}$ be the set of ideal points of $H$ and let
$\mathcal{L}_{H}$ be the space of those lines in $H^{n}$ whose two ideal
points are in different components of $\partial B^{n}\setminus I$. Note that
$\mathcal{L}_{H}$ is locally compact, $\ell\in\mathcal{L}_{H}$ and we can
assume that $\ell_{m}\in\mathcal{L}_{H}$ for all $m$. Define

$f_{H}\colon\mathcal{L}_{H}\times K\to{\mathbb{R}}$

in a point $(k,x)$ in the following way: let $y$ be the intersection point of
the line $k$ with $H$ and let $z$ be the point on $k$ closest to $x$ (that is,
the intersection of $k$ with the hyperplane through $x$ orthogonal to $k$);
then we put $f_{H}(k,x):=d(y,z)$ if $z$ is in the component of $H^{n}\setminus
H$ whose closure contains the ideal point $i$ and $f_{H}(k,x):=-d(y,z)$
otherwise. Now clearly

$d_{H^{n}}(a,a^{\prime})=\max_{x\in K}f_{H}(\ell,x)-\min_{x\in
K}f_{H}(\ell,x)\quad\text{and}\quad d_{H^{n}}(a_{m},a^{\prime}_{m})=\max_{x\in
K}f_{H}(\ell_{m},x)-\min_{x\in K}f_{H}(\ell_{m},x),$

hence we only need to prove that $\max f_{H}(\ell_{m},\cdot)\to\max
f_{H}(\ell,\cdot)$ and $\min f_{H}(\ell_{m},\cdot)\to\min f_{H}(\ell,\cdot)$.
This will immediately follow from lemma 5.1 as soon as we prove that $f_{H}$
is continuous.

Let $(k_{m},x_{m})\to(k,x)$ be a convergent sequence of lines in
$\mathcal{L}_{H}$ and points in $K$, moreover, let $y,z$ be the points on $k$
whose distance defines the value of $f_{H}(k,x)$ as above and let
$y_{m},z_{m}$ be their analogues on $k_{m}$ defining $f_{H}(k_{m},x_{m})$. It
suffices to prove the convergences $y_{m}\to y$ and $z_{m}\to z$. For the
proof of $y_{m}\to y$ we use the Cayley–Beltrami–Klein model where hyperbolic
lines and hyperplanes are represented as the intersections of
$\mathrm{int}B^{n}$ with Euclidean lines and hyperplanes respectively. Clearly
the convergence of ideal points (and so of lines) is the same as in the
Poincaré disk model and now the points $y_{m}$ and $y$ are realised as
intersections of a Euclidean hyperplane with Euclidean line segments whose
endpoints converge, hence the convergence $y_{m}\to y$ follows. For the proof
of $z_{m}\to z$ let $z^{\prime}_{m}$ denote the point on $k_{m}$ closest to
$x$. Choosing a Cayley–Beltrami–Klein model where $x$ is the centre of $B^{n}$
yields that $z^{\prime}_{m}$ is represented as the intersection of $k_{m}$
with the Euclidean hyperplane through $x$ and orthogonal to $k_{m}$, hence we
get $z^{\prime}_{m}\to z$. Since $x_{m}\to x$, for any $\varepsilon>0$ we have
$|d(x,z^{\prime}_{m})-d(x_{m},z^{\prime}_{m})|<\varepsilon$ and
$|d(k_{m},x)-d(k_{m},x_{m})|<\varepsilon$ for sufficiently large indices $m$,
which implies

$|d(x_{m},z_{m})-d(x_{m},z^{\prime}_{m})|\leq|d(x_{m},z_{m})-d(x,z^{\prime}_{m})|+|d(x,z^{\prime}_{m})-d(x_{m},z^{\prime}_{m})|<2\varepsilon.$

Thus $|d(z_{m},z^{\prime}_{m})|\to 0$ and so $z_{m}\to z$ which finishes the
proof. ∎

######  Proposition 5.3.

The Santaló width $w_{S}(K,(z,H_{z}))$ is a continuous function of
$(K,(z,H_{z}))$.

###### Proof.

Let $K_{m}\to K$ be a convergent sequence of convex bodies and $H_{m}$
supporting hyperplanes through $z_{m}\in\partial K_{m}$ such that
$(z_{m},H_{m})\to(z,H_{z})$. If $\ell_{m}$ and $\ell$ denote the lines through
$z_{m}$ and $z$ orthogonal to $H_{m}$ and $H$ respectively, then it is not
hard to see that $\ell_{m}\to\ell$. Now the previous lemma implies that we
have $w_{S}(K_{m},(z_{m},H_{m}))\to w_{S}(K,(z,H_{z}))$, hence $w_{S}$ is
indeed continuous. ∎

######  Lemma 5.4.

If $K\subset H^{n}$ is a convex body, $i$ an ideal point and $i_{m}$ a
sequence of ideal points with $i_{m}\to i$, then we have $w_{F}(K,i_{m})\to
w_{F}(K,i)$.

###### Proof.

Let $o$ be the centre of the disk $B^{n}$ (where again $\mathrm{int}B^{n}$ is
identified with $H^{n}$). We define the value of

$h_{o}\colon\partial B^{n}\times K\to{\mathbb{R}}$

at a point $(j,x)$ as follows: let $y$ be the intersection of the horosphere
at $j$ through $x$ with the geodesic line through $o$ tending to $j$ oriented
positive from $o$ in the direction of $j$ and negative in the direction of
$-j$; then $h_{o}(j,x)$ is the oriented distance of $y$ from $o$. Now we have

$w_{F}(K,j)=\max_{x\in K}h_{o}(j,x)-\min_{x\in K}h_{o}(j,x),$

hence we only need to prove that $\max h_{o}(i_{m},\cdot)$ and $\min
h_{o}(i_{m},\cdot)$ converge to $\max h_{o}(i,\cdot)$ and $\min
h_{o}(i,\cdot)$ respectively. Since $h_{o}$ is clearly continuous, we can now
apply lemma 5.1 with $X=\partial B^{n}$, $Y=K$ and $f=h_{o}$ for the maximum
and $f=-h_{o}$ for the minimum. ∎

######  Proposition 5.5.

The Fillmore width $w_{F}(K,i)$ is a continuous function of $(K,i)$.

###### Proof.

Let $(K_{m},i_{m})\to(K,i)$ be a convergent sequence of pairs of convex bodies
and ideal points and fix an $\varepsilon>0$. If $m$ is large enough, then
$K_{m}$ and $K$ are contained in the convex bodies $K^{(\varepsilon)}$ and
$K_{m}^{(\varepsilon)}$ respectively, hence for any ideal point $j$ we have

$w_{F}(K,j)-2\varepsilon\leq w_{F}(K_{m}^{(\varepsilon)},j)-2\varepsilon\leq
w_{F}(K_{m},j)\leq w_{F}(K^{(\varepsilon)},j)\leq w_{F}(K,j)+2\varepsilon$

Now the previous lemma implies that for $m$ sufficiently large we have
$|w_{F}(K,i_{m})-w_{F}(K,i)|<\varepsilon$, thus we obtain

$|w_{F}(K_{m},i_{m})-w_{F}(K,i)|\leq|w_{F}(K_{m},i_{m})-w_{F}(K,i_{m})|+|w_{F}(K,i_{m})-w_{F}(K,i)|<5\varepsilon$

for all suitably large indexes $m$ which finishes the proof. ∎

######  Lemma 5.6.

If $K\subset H^{n}$ is a convex body, $H$ a hyperplane intersecting $K$ and
$H_{m}$ a sequence of hyperplanes intersecting $K$ such that $H_{m}\to H$,
then we have $w(K,H_{m})\to w(K,H)$.

###### Proof.

Let $I\subset\partial B^{n}$ be the set of ideal points of $H$ in a Poincaré
disk model and let $c,-c$ be the two (antipodal) centre points of $I$ on the
sphere. Denote by $\mathcal{H}_{H}$ the space of those hyperplanes
$H^{\prime}\subset H^{n}$ whose analogous two centre points are in different
components of $\partial B^{n}\setminus I$. Observe that $\mathcal{H}_{H}$ is
locally compact, $H\in\mathcal{H}_{H}$ and we can also assume that
$\mathcal{H}_{H}$ contains $H_{m}$ for all $m$. We define the value of

$g_{H}\colon\mathcal{H}_{H}\times K\to{\mathbb{R}}$

in a hyperplane $H^{\prime}$ and point $x$ as the distance of $x$ and
$H^{\prime}$ with positive sign if $x$ is in the half-space whose closure
contains $-c$ and with negative sign otherwise. Then we have

$w(K,H^{\prime})=\max_{x\in K}g_{H}(H^{\prime},x)-\min_{x\in
K}g_{H}(H^{\prime},x)$

for any hyperplane $H^{\prime}$ intersecting $K$, thus it suffices to prove
the convergences $\max g_{H}(H_{m},\cdot)\to\max g_{H}(H,\cdot)$ and $\min
g_{H}(H_{m},\cdot)\to\min g_{H}(H,\cdot)$. Now we can apply lemma 5.1 with
$f=g_{H}$ for the maximum and $f=-g_{H}$ for the minimum provided that $g_{H}$
is continuous.

If we have a convergent sequence of hyperplanes $H^{\prime}_{m}\to H^{\prime}$
in $\mathcal{H}_{H}$ and a convergent sequence of points $x_{m}\to x$ in $K$,
then we need that the distances $d(H^{\prime}_{m},x_{m})$ converge to
$d(H^{\prime},x)$. This follows analogously to the way we obtained the
convergence $z_{m}\to z$ in the proof of lemma 5.2 so we omit its proof here.
∎

Now a word-by-word analogue of the proof of proposition 5.5 yields the
following.

######  Proposition 5.7.

The extended Leichtweiss width $w(K,H)$ is a continuous function of $(K,H)$.

As a special case of this we also get the following.

######  Proposition 5.8.

The Leichtweiss width $w_{L}(K,(p,H_{p}))$ is a continuous function of
$(K,(p,H_{p}))$.

######  Lemma 5.9.

Let $z_{m}\to z$ and $z^{\prime}_{m}\to z^{\prime}$ be convergent sequences of
points in $H^{2}$ and let $\ell_{m}\to\ell_{z}$ and
$\ell^{\prime}_{m}\to\ell_{z^{\prime}}$ be convergent sequences of lines such
that $z_{m}\in\ell_{m}$, $z\in\ell_{z}$, $z^{\prime}_{m}\in\ell^{\prime}_{m}$
and $z^{\prime}\in\ell_{z^{\prime}}$, the sequence $\ell_{m}$ converges to
$\ell_{z}$ and $\ell_{m}$ and $\ell^{\prime}_{m}$ (resp. $\ell_{z}$ and
$\ell_{z^{\prime}}$) have equal alternating angles with the segment
$[z_{m},z^{\prime}_{m}]$ (resp. $[z,z^{\prime}]$). Then we have
$d(\ell_{m},\ell^{\prime}_{m})\to d(\ell_{z},\ell_{z^{\prime}})$.

###### Proof.

Since the distance of lines is continuous, it is sufficient to prove that
$\ell^{\prime}_{m}\to\ell_{z^{\prime}}$. Let $\varphi_{m}$ denote the angle of
the line $\ell_{m}$ and the segment $[z_{m},z^{\prime}_{m}]$ which is oriented
positive from the line to the segment; similarly let $\varphi$ be the positive
angle from $\ell_{z}$ to $[z,z^{\prime}]$. Observe that $\varphi_{m}$ tends to
$\varphi$ and the positive angle from the line $\ell^{\prime}_{m}$ (resp.
$\ell_{z^{\prime}}$) to the segment $[z^{\prime}_{m},z_{m}]$ (resp.
$[z^{\prime},z]$) is again $\varphi_{m}$ (resp. $\varphi$). Now if
$\ell^{\prime\prime}_{m}$ is the line through $z^{\prime}_{m}$ which has
positive angle $\varphi$ with $[z^{\prime}_{m},z_{m}]$, then it is not hard to
see the convergence $\ell^{\prime\prime}_{m}\to\ell^{\prime}_{z}$. Let
$i^{\prime}_{m}$, $j^{\prime}_{m}$ and $i^{\prime\prime}_{m}$,
$j^{\prime\prime}_{m}$ be the ideal points of $\ell^{\prime}_{m}$ and
$\ell^{\prime\prime}_{m}$ respectively such that $\varphi_{m}$ is the angle
$\angle(i^{\prime}_{m},z^{\prime}_{m},z_{m})$ and
$\varphi=\angle(i^{\prime\prime}_{m},z^{\prime}_{m},z_{m})$. Now the angle of
$\ell^{\prime}_{m}$ and $\ell^{\prime\prime}_{m}$ tends to $0$, hence if
$B^{2}$ denotes the disk of a Poincaré disk model we use, the sequence of the
(Euclidean) distances $d_{\partial
B^{2}}(i^{\prime}_{m},i^{\prime\prime}_{m})$ and $d_{\partial
B^{2}}(j^{\prime\prime}_{m},j^{\prime}_{m})$ of the corresponding ideal points
both tend to zero. This implies that $\ell^{\prime}_{m}$ also converges to
$\ell_{z^{\prime}}$ which concludes the proof. ∎

######  Proposition 5.10.

The Jerónimo-Castro–Jimenez-Lopez width $w_{JCJL}(K,(z,\ell_{z},z^{\prime}))$
is a continuous function of $(K,(z,\ell_{z},z^{\prime}))$.

###### Proof.

Let $K_{m}\to K$ be a convergent sequence of convex bodies and let
$z_{m},z^{\prime}_{m}\in\partial K_{m}$ two sequences of points converging to
$z,z^{\prime}\in\partial K$ respectively. Let $\ell_{m}\to\ell_{z}$ be a
convergent sequence of lines such that $\ell_{m}$ (resp. $\ell_{z}$) is a
supporting line of $K_{m}$ at $z_{m}$ (resp. of $K$ at $z$) and suppose that
the point $z^{\prime}_{m}$ (resp. $z^{\prime}$) is an opposite point to
$z_{m}$ (resp. $z$) with respect to $\ell_{m}$ (resp. $\ell$). Denote by
$\ell^{\prime}_{m}$ the supporting line through $z^{\prime}_{m}$ such that
$\ell^{\prime}_{m}$ and $\ell_{m}$ have equal alternating angles with
$[z_{m},z^{\prime}_{m}]$ and let $\ell_{z^{\prime}}$ be the analogous
supporting line through $z^{\prime}$. Now since we have
$w_{JCJL}(K_{m},(z_{m},\ell_{m},z^{\prime}_{m}))=d(\ell_{m},\ell^{\prime}_{m})$
and $w_{JCJL}(K,(z,\ell_{z},z^{\prime}))=d(\ell_{z},\ell_{z^{\prime}})$, the
previous lemma yields that $w_{JCJL}(K_{m},(z_{m},\ell_{m},z^{\prime}_{m}))\to
w_{JCJL}(K,(z,\ell_{z},z^{\prime}))$. This shows that
$w_{JCJL}(K,(z,\ell_{z},z^{\prime}))$ is indeed continuous. ∎

######  Lemma 5.11.

If $K\subset H^{2}$ is a convex body and $i_{m}\to i$ is a convergent sequence
of ideal points, then we have $w_{GH}(K,i_{m})\to w_{GH}(K,i)$.

###### Proof.

Recall that we have $w_{GH}(K,j)=\max\\{w(K,\ell)\colon j\text{ is an ideal
point of }\ell\\}$. Now putting $v(j,x):=w(K,\ell_{j,x})$ where $\ell_{j,x}$
is the line with ideal point $j$ through $x$, proposition 5.7 implies that $v$
is continuous and by design we get

$w_{GH}(K,j)=\max_{x\in K}v(j,x).$

Now we can apply lemma 5.1 with $X=\partial B^{n}$ and $Y=K$ which yields that
$w_{GH}(K,j)$ is continuous in the variable $j$, thus our claim follows. ∎

Now again a direct analogue of the proof of proposition 5.5 implies the
following.

######  Proposition 5.12.

The G. Horváth width $w_{GH}(K,i)$ is a continuous function of $(K,i)$.

Remark  It is worthwhile to note that in the proofs above we never used the
convexity of the bodies $K$ and $K_{m}$, hence the support functions $f_{H}$,
$g_{H}$ and $h_{o}$ defined in the proofs of the lemmas 5.2, 5.4 and 5.6 can
be used to extend these notions of width to any compact subset of $H^{n}$ (and
with these definitions the width of any compact subset will be the width of
its convex hull). The width functions will then be continuous in this extended
sense too.

We finish this section with showing that the minimal width functions defined
by all of the above widths are continuous as well.

######  Theorem 5.13.

If $K_{m}\to K$ is a convergent sequence of convex bodies in $H^{n}$, then

(i):

$w_{S}(K_{m})\to w_{S}(K)$;

(ii):

$w_{F}(K_{m})\to w_{F}(K)$;

(iii):

if $p_{m}\in K_{m}$ and $p\in K$ are interior points and $p_{m}\to p$, then
$w_{L}(K_{m},p_{m})\to w_{L}(K,p)$;

(iv):

if $n=2$, then $w_{JCJL}(K_{m})\to w_{JCJL}(K)$;

(v):

if $n=2$, then $w_{GH}(K_{m})\to w_{GH}(K)$;

(vi):

$w(K_{m})\to w(K)$.

###### Proof.

As we saw above, all of these widths are continuous as functions of $K$ and a
second object, which is either an ideal point or a hyperplane (or line)
intersecting or supporting $K$ occasionally with fixed additional points. We
want to apply lemma 5.1 to the negative of these width functions with $X$
being the space of compact subsets and $Y$ the space of the objects in the
second variable in all cases. The Blaschke selection theorem 3.4 implies that
this $X$ is locally compact, hence if $Y$ is the space of ideal points (which
is compact), i.e. for $w_{F}$ and $w_{GH}$, then the lemma may indeed be
applicable. To be able to apply the lemma in the other cases, note that it is
sufficient to use it with $X$ being only the space of compact subsets of
$K^{(\varepsilon)}$ for some $\varepsilon>0$. Then we can consider only those
hyperplanes which intersect $K^{(\varepsilon)}$, which form a closed subspace

$\mathcal{H}_{K,\varepsilon}\subset{\raisebox{1.99997pt}{$(\partial
B^{n}\times[\delta,\pi-\delta])$}\left/\raisebox{-1.99997pt}{$(c,r)\sim(-c,\pi-r)$}\right.}\subset\mathcal{H}$

for some $\delta>0$, hence it is compact. Now for the width functions $w_{L}$
and $w$ we can put $Y:=\mathcal{H}_{K,\varepsilon}$ and for $w_{S}$ and
$w_{JCJL}$ the product spaces $\mathcal{H}_{K,\varepsilon}\times
K^{(\varepsilon)}$ and
$K^{(\varepsilon)}\times\mathcal{H}_{K,\varepsilon}\times K^{(\varepsilon)}$
respectively. Then we can define for each convex body $K^{\prime}$ the
subspace $Z_{K^{\prime}}\subset Y$ as the set of those objects where the
appropriate width function is defined, which is closed in all cases. Hence
lemma 5.1 can be applied and our statement follows. ∎

This finishes the proof of theorem 1.1.

##  6\. Width and diameter

In what follows we will consider the maximal widths defined by the width
functions $w_{S}$, $w_{F}$, $w_{L}$, $w_{JCJL}$, $w_{GH}$ and $w$ and we shall
see that this always gives the diameter

######  Lemma 6.1.

For a convex body $K\subset H^{n}$,

$\operatorname{\mathrm{diam}}K=\max\left\\{w_{S}\left(K,\left(z,H_{z}\right)\right)\colon
z\in\partial K\text{ and }H_{z}\text{ is a supporting hyperplane of }K\text{
at }z\right\\}.$

###### Proof.

Let $z\in\partial K$ an arbitrary boundary point and $H_{z}$ any supporting
hyperplane to $K$, such that $z\in H_{z}$. If $\ell$ is the unique hyperbolic
line orthogonal to $H_{z}$ through the point $z$, and $H^{\prime}_{z}$ is the
other supporting hyperplane to $K$ orthogonal to $\ell$, intersecting $\ell$
at the point $z^{\prime}$, then there is some point $z_{0}\in
H^{\prime}_{z}\cap\partial K$. Because of the orthogonality, and from the
definition of the diameter,

$w_{S}\left(K,\left(z,H_{z}\right)\right)=d\left(z,z^{\prime}\right)\leq
d\left(z,z_{0}\right)\leq\operatorname{\mathrm{diam}}K$

follows. On the other hand, if $\left[z,z^{\prime}\right]$ is a diametral
chord in $K$, then the hyperplanes $H_{z}$ and $H_{z^{\prime}}$ through the
points $z$ and $z^{\prime}$ respectively and both orthogonal to the geodesic
line connecting $z$ and $z^{\prime}$ are supporting $K$ at $z$ and
$z^{\prime}$, respectively. Otherwise, if there is a point, say $z_{0}\in K$
in the open half-space bounded by $H_{z^{\prime}}$ not containing $H_{z}$,
then

$d\left(z,z_{0}\right)>d\left(z,z^{\prime}\right)=\operatorname{\mathrm{diam}}K$

is a contradiction. Therefore, in this case

$w_{S}\left(K,\left(z,H_{z}\right)\right)=w_{S}\left(K,\left(z^{\prime},H_{z^{\prime}}\right)\right)=d\left(z,z^{\prime}\right)=\operatorname{\mathrm{diam}}K.$

This concludes the proof. ∎

######  Lemma 6.2.

Let $K\subset H^{n}$. Then,

$\operatorname{\mathrm{diam}}K=\max\left\\{w_{F}\left(K,i\right)\colon i\text{
is an ideal point}\right\\}.$

###### Proof.

Let $i$ be any ideal point, and let $\partial A$ and $\partial C$ be the two
parallel supporting horospheres to $K$ at the ideal point $i$. Then, there is
a point $x\in\partial A\cap\partial K$ and $y\in\partial C\cap\partial C$. Let
$z$ be the intersection of the horosphere $\partial C$ and the line through
the ideal point $i$ and $x$, orthogonal to both of these supporting
horospheres. Then,

$\operatorname{\mathrm{diam}}K\geq d\left(x,y\right)\geq
d\left(x,z\right)=w_{F}\left(K,i\right).$

On the other hand, if $\left[x,y\right]\subset K$ is a diametral chord, and
$i$ is one of the ideal points determined by the geodesic line $\ell$ through
$x$ and $y$, then if $\partial A$ is the orthogonal horosphere to $i$ through
$x$ and $\partial C$ is the orthogonal horosphere to $i$ containing $y$,
$\partial A$ and $\partial C$ need to be supporting horospheres of $K$.
Otherwise, if there is a point say $z\in\text{\rm int }A\cap K$, where we
assume $A\subset C$, then

$d\left(y,z\right)\geq
d\left(x_{0},z\right)+d\left(x_{0},y_{0}\right)>d\left(x_{0},y_{0}\right)=\operatorname{\mathrm{diam}}K$

is a contradiction, where $x_{0}$ is the intersection point of the line
$\ell_{z}$ through $i$ and $z$ with the horosphere $\partial A$ and $y_{0}$ is
the intersection of $\ell_{z}$ and $\partial C$. ∎

######  Lemma 6.3.

Let $K\subset H^{n}$ be a convex body and $p\in\text{\rm int }K$ a fixed
interior point. Then,

$\operatorname{\mathrm{diam}}K=\max\left\\{w_{L}\left(K,\left(p,H\right)\right)\colon
H\text{ is a hyperplane containing }p\right\\}.$

###### Proof.

Let $H$ be any hyperplane containing $p$, and $\mathcal{A}$ be the
hyperspherical domain with respect to $H$ with boundary components $H_{1}$ and
$H_{2}$. Then, there are some points $x_{1}\in H_{1}\cap\partial K$ and
$x_{2}\in H_{2}\cap\partial K$. Let $x$ be the intersection point of the
geodesic segment $\left[x_{1},x_{2}\right]$ and the hyperplane $H$, and let
$y_{1}$ and $y_{2}$ be the orthogonal projection of $x_{1}$ and $x_{2}$ onto
$H$, respectively. Then,

$\operatorname{\mathrm{diam}}K\geq
d\left(x_{1},x_{2}\right)=d\left(x_{1},x\right)+d\left(x_{2},x\right)\geq
d\left(x_{1},y_{1}\right)+d\left(x_{2},y_{2}\right)=w_{L}\left(K,\left(p,H\right)\right).$

On the other hand, if $\left[x_{1},x_{2}\right]\subset K$ is a diametral
chord, then the hyperplane $H$ containing $p$ and orthogonal to
$\left[x_{1},x_{2}\right]$ determines a hyperspherical domain $\mathcal{A}$ of
boundary components $H_{1}\ni x_{1}$ and $H_{2}\ni x_{2}$. This has to be a
supporting hyperspherical domain, otherwise if there is a point $y\in\text{\rm
int }K\setminus\mathcal{A}$ (say in $H_{1}^{-}$) and $\ell$ denotes the line
through $y$ and orthogonal to $H$, while $y_{1}$ and $y_{2}$ are the
intersection points $\ell\cap H_{1}$ and $\ell\cap H_{2}$, then

$d\left(y,x_{2}\right)\geq
d\left(y,y_{2}\right)=d\left(y,y_{1}\right)+d\left(y_{1},y_{2}\right)=d\left(y,y_{1}\right)+d\left(x_{1},x_{2}\right)>d\left(x_{1},x_{2}\right)=\operatorname{\mathrm{diam}}K$

is a contradiction. ∎

######  Lemma 6.4.

For a convex body $K\subset H^{n}$,

$\operatorname{\mathrm{diam}}K=\max\left\\{w\left(K,H\right)\colon H\text{ is
a hyperplane intersecting }K\right\\}.$

###### Proof.

If there is an interior point $p\in\text{\rm int }K\cap H$, then we know from
Lemma 6.3 that $w\left(K\right)\leq\operatorname{\mathrm{diam}}K$. Similarly,
if $H$ is a supporting hyperplane to $K$ and $p\in\partial K\cap H$, $\ell$ is
the geodesic line through $p$ orthogonal to $H$ then for a sequence of
interior points $\left\\{p_{m}\right\\}_{m\in{\mathbb{N}}}\subset\text{\rm int
}K\cap\ell$ such that $\lim_{m\to\infty}p_{m}=p$ and hyperplanes $H_{m}$
orthogonal to $\ell$ through the point $p_{m}$, from Proposition 5.7, we can
deduce

$w\left(K,H\right)\leq\limsup_{m\in{\mathbb{N}}}w\left(K,H_{m}\right)\leq\operatorname{\mathrm{diam}}K.$

On the other hand, if $\left[z_{1},z_{2}\right]\subset K$ is a diametral
chord, $\ell$ denotes the geodesic line through $z_{1}$ and $z_{2}$, then as
we have seen in Lemma 6.3, that for any hyperplane $H$ orthogonal to $\ell$
and intersecting $K$ in some interior point,
$w\left(K,H\right)=\operatorname{\mathrm{diam}}K$. Moreover, if $H$ is a
supporting hyperplane orthogonal to $\ell$ to $K$ (say at $z_{1}$), then for
some interior points
$\left\\{p_{m}\right\\}_{m\in{\mathbb{N}}}\subset\text{\rm int
}\left[z_{1},z_{2}\right]$ such that $\lim_{m\to\infty}p_{m}=z_{1}$ and
orthogonal hyperplanes $H_{m}$ to $\ell$ through $p_{m}$, as
$w\left(K,H_{m}\right)=\operatorname{\mathrm{diam}}K$ for all
$m\in{\mathbb{N}}$, Proposition 5.7 implies

$w\left(K,H\right)=\lim_{m\to\infty}w\left(K,H_{m}\right)=\operatorname{\mathrm{diam}}K.$

∎

Remark  For $n=2$ we have

$\max\left\\{w_{GH}\left(K,i\right)\colon i\text{ ideal
point}\right\\}=\max\left\\{w\left(K,\ell\right)\colon\ell\text{ is a line
intersecting }K\right\\},$

so in this case the maximum of the G. Horváth width also equals to the
diameter by the above lemma.

G. Horváth proved[7] the following comparison of the Santaló width and the
Jerónimo-Castro–Jimenez-Lopez width on the hyperbolic plane.

######  Lemma 6.5.

Let $K\subset H^{2}$ be a convex body, let $z\in\partial K$ be any boundary
point, $\ell_{z}$ a supporting hyperplane at $z$ to $K$ and $z^{\prime}$ an
opposite point to $z$ with respect to $\ell_{z}$. Then

$w_{JCJL}\left(K,\left(z,\ell_{z},z^{\prime}\right)\right)\geq
w_{S}\left(K,\left(z,\ell_{z}\right)\right).$

This and Lemma 6.1 implies the following.

######  Corollary 6.6.

Let $K\subset H^{2}$ a convex body. Then,

$\operatorname{\mathrm{diam}}K=\max\left\\{w_{JCJL}\left(K,\left(z,\ell_{z},z^{\prime}\right)\right)\colon
z,z^{\prime}\in\partial K\text{ are opposite with respect to
}\ell_{z}\right\\}.$

Remark  A similar reasoning as in the proof of Theorem 5.13 also yields the
statement of Lemma 3.5, i.e. that the diameter (which is the maximal width
instead of the minimal) is continuous.

##  7\. Bodies of constant width in the hyperbolic space

We call a $K\subseteq H^{n}$ set _complete_ if for each $y\in H^{n}\setminus
K$,
$\operatorname{\mathrm{diam}}\left(K\cup\left\\{y\right\\}\right)>\operatorname{\mathrm{diam}}\left(K\right)$.
Readily, any complete set is closed. A complete set $\widetilde{K}$ is called
a _completion_ of $K$ if $K\subseteq\widetilde{K}$ and they have the same
diameter. The Zorn lemma already yields the existence of completions (which is
in general not unique, see Dekster[4]), but as Theorem 1.3 will show, we can
also give a construction of a completion, such that the completion has the
same circumscribed ball as the original body.

For the notion of bodies of constant width, we use Dekster’s definition
(see[4]). A convex body $K\subset H^{n}$ is _of constant width_ $D>0$, if for
any boundary point $x\in\partial K$ and any outer normal unit vector $v\in
T_{x}$, there is a boundary point $y\in\partial K$ such that
$\left[x,y\right]\subset K$ is of length $D$ and $-v$ points along the
geodesic segment $\left[x,y\right]$. We give the following characterization
for bodies of constant width in $H^{n}$.

Theorem 1.2 Let $K\subset H^{n}$ be a convex body of diameter $D>0$. The
following are equivalent:

(i):

$K$ is complete;

(ii):

$K=\bigcap_{x\in K}B\left(x,D\right)$;

(iii):

$K$ is of constant width $D$;

(iv):

$w_{S}\left(K,\left(z,H_{z}\right)\right)=D$ for all $z\in\partial K$ and
$H_{z}$ supporting hyperplanes to $K$ at $z$;

(v):

$w_{F}\left(K,i\right)=D$ for all ideal points $i$;

(vi):

$w_{L}\left(K,\left(p,H_{p}\right)\right)=D$ for all hyperplanes through a
fixed inner point $p\in\text{\rm int }K$;

(vii):

$w\left(K,H\right)=D$ for all hyperplanes $H$ intersecting $K$.

###### Proof.

For the equivalence of (i), (ii), (iii) and (v), see Böröczky, Sagmeister[2].

Assume that $K$ is of constant width $D$ and let $z\in\partial K$ be any
boundary point and $H_{z}$ any supporting hyperplane at $z$ to $K$. Let $\ell$
be the geodesic line containing $z$ orthogonal to $H_{z}$, $z^{\prime}$ will
be the intersection point of $\ell$ and $\partial
K\setminus\left\\{z\right\\}$, while $v\in T_{z}$ is the outer normal unit
vector, such that $-v$ points along the geodesic segment
$\left[z,z^{\prime}\right]$. Then, since $K$ is of constant width, the
geodesic segment $\left[z,z^{\prime}\right]$ is a diametral chord, so as we
have seen in Lemma 6.1, the hyperplane $H_{z^{\prime}}$ containing
$z^{\prime}$ and orthogonal to $\ell$ supports $K$, therefore
$w_{S}\left(K,\left(z,H_{z}\right)\right)=D$, hence (iii) implies (iv).

Now we assume that $w_{S}\left(K,\left(z,H_{z}\right)\right)=D$ for all
boundary point $z\in\partial K$ and $H_{z}$ supporting hyperplane of $K$ at
$z$. Then from Lemma 6.1, $D=\operatorname{\mathrm{diam}}K$. Suppose for
contradiction that $K$ is not complete. Let $\widetilde{K}\supsetneq K$ be a
completion of $K$, and $y\in\widetilde{K}\setminus K$. As $K$ is a convex
body, there is a supporting hyperplane $H$ to $K$ separating $y$ and $K$. But
then, if $z_{0}\in H\cap\partial K$, since
$w_{S}\left(K,\left(z_{0},H\right)\right)=D$, there is a
$z^{\prime}\in\partial K$ and a $H^{\prime}$ supporting hyperplane to $K$ at
$z^{\prime}$, such that $\left[z_{0},z^{\prime}\right]$ is a diametral chord
orthogonal both to $H$ and $H^{\prime}$. But then

$D=\operatorname{\mathrm{diam}}\widetilde{K}\geq
w_{S}\left(\widetilde{K},z^{\prime}\right)>D=w_{S}\left(K,z^{\prime}\right)$

is a contradiction, therefore $K$ has to be complete. So (iv) implies (i).

We again assume that $K$ is of constant width $D$. Let $H$ be any hyperplane
intersecting $K$. Let $A$ and $C$ be the two supporting hyperspheres
determined by $H$ (where in the degenerate case one of them is the supporting
hyperplane $H$), and let $z\in A\cap\partial K$. If $\ell_{z}$ is the geodesic
line through $z$ and orthogonal to $H$, then the hyperplane $H_{z}\ni z$
orthogonal to $\ell_{z}$ supports $K$ at $z$. Let $z^{\prime}$ be the
intersection point of $\ell_{z}$ and $\partial K\setminus\left\\{z\right\\}$.
If $v\in T_{z}$ is the outer normal unit vector, such that $-v$ points along
the geodesic segment $\left[z,z^{\prime}\right]$, then
$d\left(z,z^{\prime}\right)=D$ by the assumption. On the other hand, by the
orthogonality of $\ell_{z}$ and $H$, $z^{\prime}\in C$ follows. Hence, (iii)
implies (vii) and (vi).

It is obvious, that (vii) implies (vi). We now assume, that for a fixed
interior point $p\in\text{\rm int }K$ each hyperplane $H$ containing $p$, the
Leichtweiss width $w_{L}\left(K,\left(p,H\right)\right)=D$. We know from Lemma
6.3, that $D=\operatorname{\mathrm{diam}}K$. Suppose for contradiction, that
$K$ is not complete, so there is a completion $\widetilde{K}\supsetneq K$ of
$K$ and a point $y\in\widetilde{K}\setminus K$. But then, there is a boundary
point $z\in\partial K$ and a hyperplane $H_{z}$ which supports $K$ at $z$,
separating $K$ and $y$. If $\ell$ is the line through $z$ and orthogonal to
$H_{z}$, and $H$ is the hyperplane containing $p$ and orthogonal to $\ell$,
then

$D=w_{L}\left(\widetilde{K},\left(p,H\right)\right)>w_{L}\left(K,\left(p,H\right)\right)=D$

is a contradiction. Therefore, $K$ is already complete, hence (vi) also
implies (ii). ∎

Additionally, for convex bodies in the hyperbolic plane, Lemma 6.5, Corollary
6.6 and Theorem 1.2 implies the following.

######  Proposition 7.1.

Let $K\subset H^{2}$ a convex body of diameter $D$. Then, $K$ is of constant
width if and only if
$w_{JCJL}\left(K,\left(z,\ell_{z},z^{\prime}\right)\right)=D$ for all
$z\in\partial K$, $\ell_{z}$ supporting line at $z$ and $z^{\prime}$ opposite
point to $z$ with respect to $\ell_{z}$

Remark However, being of constant width $D$ is not equivalent with being of
constant G. Horváth width in $H^{2}$. As G. Horváth pointed out in his
paper[7], there are regular $\left(2k+1\right)$-gons of constant G. Horváth
width, which are not complete (a completion of theirs are the well-known
Reuleaux polygons).

We deduce the following properties of complete sets from Proposition 1.2 and
Lemma 3.1.

######  Corollary 7.2.

Let $K\subset H^{n}$ a $D$-complete set where $D>0$. Then the following hold:

  1. (1)

$K$ is convex and compact

  2. (2)

For all $z\in\partial K$ there is a $y\in\partial K$ such that
$d\left(y,z\right)=D$

  3. (3)

For for any pair of points $y,z\in K$ if $\sigma$ denotes the shorter arc of
some circle of radius at least $D$ through $y$ and $z$, then $\sigma\subseteq
K$

  4. (4)

For any pair of points $y,z\in K$ if a horocycle arc $\theta$ connects $y$ and
$z$, then $\theta\subseteq K$.

##  8\. The circumscribed ball and inscribed balls of convex bodies

For a compact convex set $K$ in $H^{n}$, we define the inradius and
circumradius of $K$ as the maximal (resp. minimal) radius of a ball contained
in (resp. containing) $K$. We use the notations $r\left(K\right)$ and
$R\left(K\right)$, so

$r\left(K\right)=\max\left\\{r\geq 0\colon\exists z\in H^{n}\colon
B\left(z,r\right)\subseteq K\right\\}$

and

$R\left(K\right)=\min\left\\{R\geq 0\colon\exists z\in H^{n}\colon K\subseteq
B\left(z,r\right)\right\\}.$

We fix a regular $d$-dimensional simplex in $H^{n}$ of edge length $D$ and
denote it by $\Delta\left(D\right)$. Note that
$\operatorname{\mathrm{diam}}\left(\Delta\left(D\right)\right)=D$.

The following theorem of Jung is well-known in $H^{n}$ (see Dekster[5]), but
we need a slightly different version. For the proof of the exact statement
below, see Böröczky, Sagmeister[2].

######  Lemma 8.1.

If $K\subset H^{n}$ a compact set of diameter $D>0$, then

(i):

there is a unique point $p$ such that $K\subset
B\left(p,R\left(K\right)\right)$;

(ii):

for $2\leq k\leq n+1$, there is a set
$\left\\{q_{1},\ldots,q_{k}\right\\}\subset\partial K\cap\partial
B\left(p,R\left(K\right)\right)$ such that
$p\in\mathrm{relint}\left[q_{1},\ldots,q_{k}\right]$ where
$\left[q_{1},\ldots,q_{k}\right]$ is a $\left(k-1\right)$-dimensional simplex
in $H^{n}$;

(iii):

$\frac{D}{2}\leq R\left(K\right)\leq R\left(\Delta\left(D\right)\right)<D;$

(iv):

$R\left(K\right)=R\left(\Delta\left(D\right)\right)$ holds if and only if
$k=n+1$ and $\left[q_{1},\ldots,q_{n+1}\right]$ is congruent with
$\Delta\left(D\right)$ in (ii).

Concerning the inradius, we have the following result. For a proof, see
Böröczky, Sagmeister[2].

######  Lemma 8.2.

If $K\subseteq H^{n}$ is a convex body, and
$B\left(w,r\left(K\right)\right)\subset K$, then for $2\leq k\leq n+1$ there
are points $t_{1},\ldots,t_{k}\in\partial K\cap\partial
B\left(w,r\left(K\right)\right)$ such that $\left[t_{1},\ldots,t_{k}\right]$
is a $\left(k-1\right)$-dimensional simplex and
$w\in{\rm{relint}}\left[t_{1},\ldots,t_{k}\right]$.

Remark Note that contrary to the circumscribed ball, the inscribed ball of a
compact convex set $K$ is not necessarily unique in the Euclidean or the
hyperbolic space. For example, take a segment $s$ and let $K$ be the set of
points of distance at most $r$ from $s$ for a fixed $r>0$.

However, for complete bodies the following holds.

######  Proposition 8.3.

If $K\subset H^{n}$ is a complete set of diameter $D>0$, then

$R\left(K\right)+r\left(K\right)=D.$

Furthermore, $K$ has a unique inscribed ball whose center is the circumcenter.

Combining Lemma 8.1 and Proposition 8.3 implies the following.

######  Corollary 8.4.

If $K\subset H^{n}$ is a complete set of diameter $D>0$, then

$R\left(K\right)\leq R\left(\Delta(D)\right)\mbox{ \ and \
}r\left(K\right)\geq D-R\left(\Delta(D)\right).$

Furthermore, either $R\left(K\right)=R\left(\Delta(D)\right)$ or
$r\left(K\right)=D-R\left(\Delta(D)\right)$ if and only if $K$ contains a
congruent copy of $\Delta(D)$.

Remark According to Zorn’s Lemma, there exists a complete set $K_{0}$ of
diameter $D$ in $H^{n}$ containing a congruent copy of $\Delta(D)$. If $n=2$,
then there exists a unique such set up to congruency; namely, the Reuleaux
triangle. For an arbitrary $n\geq 2$ the proof of the following theorem gives
a construction of such a set.

The following is well-known by Scott [14] in the Euclidean case, however, the
Euclidean proof can be easily extended to $H^{n}$, and since the original
paper is hard to access, we present the proof below.

Theorem 1.3. _Let $K\subset H^{n}$ be a convex body of diameter $D$ and
circumscribed ball $B$. Then, there exists a completion $\widetilde{K}$ of $K$
such that $B$ is also the circumscribed ball of $\widetilde{K}$._

###### Proof.

Let

$U\left(K\right)=\left\\{x\in
B\colon\operatorname{\mathrm{diam}}\left(K\cup\left\\{x\right\\}\right)=D\right\\}$

and

$\varrho\left(K\right)=\sup\left\\{d\left(x,K\right)\colon x\in
U\left(K\right)\right\\}.$

Since $U\left(K\right)\subseteq B$, $U\left(K\right)$ is obviously bounded,
and it is also easy to see that it is closed: if
$\left(y_{j}\right)_{j\in{\mathbb{N}}}\subseteq U\left(K\right)$ is a
convergent sequence and $y$ is its limit, by the compactness of $B$ we have
$y\in B$ and

$D\leq\operatorname{\mathrm{diam}}\left(K\cup\left\\{y\right\\}\right)\leq\limsup_{j\in{\mathbb{N}}}\operatorname{\mathrm{diam}}\left(K\cup\left\\{y_{j}\right\\}\right)=D,$

therefore $y\in U\left(K\right)$. This makes $U\left(K\right)$ compact, and
hence

$\varrho\left(K\right)=\max\left\\{d\left(x,K\right)\colon x\in
U\left(K\right)\right\\}.$

Let $x_{1}\in U\left(K\right)$ such that
$d\left(x_{1},K\right)=\varrho\left(K\right)$, and let
$K_{1}=\operatorname{\mathrm{conv}}\left(K\cup\left\\{x_{1}\right\\}\right)$.
We know that
$\operatorname{\mathrm{diam}}\left(K\cup\left\\{x_{1}\right\\}\right)=D$.
Furthermore $\operatorname{\mathrm{diam}}K_{1}=D$, since taking the convex
hull preserves the diameter (see e.g. Böröczky, Sagmeister [1] for a
reference). Also, $B$ is convex, so $K_{1}\subseteq B$, and hence we can
define $U\left(K_{1}\right)$ the same way as earlier. Again, this will be a
compact set in $B$, and we can define $\varrho\left(K_{1}\right)$ similarly as
before.

For $j\in{\mathbb{N}}$, $x_{j+1}$ will be a point in $U\left(K_{j}\right)$,
such that $d\left(x_{j+1},K\right)=\varrho\left(K_{j}\right)$ and let
$K_{j+1}=\operatorname{\mathrm{conv}}\left(K_{j}\cup\left\\{x_{j+1}\right\\}\right)$.
Now $\left(K_{j}\right)_{j\in{\mathbb{N}}}$ is a bounded sequence of convex
bodies in $H^{n}$ such that $K_{j}\subseteq K_{j+1}\subseteq B$ for all
$j\in{\mathbb{N}}$, therefore there is a unique convex body
$\widetilde{K}\subseteq B$, such that $\widetilde{K}=\lim_{j\to\infty}K_{j}$.
Since $\operatorname{\mathrm{diam}}K_{j}=D$ for all $j\in{\mathbb{N}}$, we
also have $\operatorname{\mathrm{diam}}\widetilde{K}=D$. It remains to show
that $\widetilde{K}$ is complete.

Let us suppose for contradiction that $\widetilde{K}$ is not complete, and let
$y\in H^{n}\setminus\widetilde{K}$ such that
$\operatorname{\mathrm{diam}}\left(K\cup\left\\{y\right\\}\right)=D$. Let
$\delta=d\left(y,\widetilde{K}\right)>0$. This also means that
$K_{j+1}\setminus K_{j}\neq\emptyset$,
$\varrho\left(K\right),\varrho\left(K_{j}\right)>0$, and $x_{j}\neq x_{j+1}$
for all $j\in{\mathbb{N}}$. We first assume that $y\in B$. Let $1\leq j<k$,
hence $K_{j}\subseteq K_{k-1}$. From the definition of $x_{j}$ and
$\varrho\left(K_{k-1}\right)$, we have that

$d\left(x_{j},x_{k}\right)\geq\varrho\left(K_{k-1}\right).$

We now show that $\varrho\left(K_{k-1}\right)\geq\delta$. Since $K\subseteq
K_{k-1}\subseteq\widetilde{K}$, $y\in B$ and
$\operatorname{\mathrm{diam}}\left(\widetilde{K}\cup\left\\{y\right\\}\right)=D$,
$\operatorname{\mathrm{diam}}\left(K_{k-1}\cup\left\\{y\right\\}\right)=D$
follows, so $y\in U\left(K_{k-1}\right)$. Also,
$K_{k-1}\subseteq\widetilde{K}$ implies $d\left(y,K_{k-1}\right)\geq\delta$.
We deduce $\varrho\left(K_{k-1}\right)\geq d\left(y,K_{k-1}\right)\geq\delta$,
and hence $d\left(x_{j},x_{k}\right)\geq\delta$, which contradicts the fact
that $\left(K_{\alpha}\right)_{\alpha\in{\mathbb{N}}}$ is convergent.
Secondly, let us assume that $y\in H^{n}\setminus B$. If $z$ is the centre of
$B$, $d\left(y,z\right)\geq R+\delta$, where $R>0$ denotes the radius of
$B=B\left(z,R\right)$. Let $z_{0}$ be the unique point on the geodesic line
through $y$ and $z$, such that $z\in\left[y,z_{0}\right]$ and
$d\left(y,z_{0}\right)=D+\delta$ (here we needed $R<D$, which holds by Lemma
8.1). It is clear from $d\left(y,z\right)=R+\delta$ and $R<D$, that
$z_{0}\not\in\widetilde{K}$, but again from Lemma 8.1 we have that
$d\left(z,z_{0}\right)=D-R\leq R$, so $z_{0}\in B\setminus\widetilde{K}$. But
then $\widetilde{K}\subset B\subset B\left(z_{0},D\right)$, so
$\operatorname{\mathrm{diam}}\left(\widetilde{K}\cup\left\\{z_{0}\right\\}\right)=D$
leads to a contradiction, as we have seen it in the previous case. This
concludes the proof. ∎

Remark Note that the same proof works in the spherical space with the geodesic
metric for complete sets of diameter less than $\frac{\pi}{2}$.

Acknowledgement:  Károly J. Böröczky and Ádám Sagmeister are supported by
NKFIH project K 132002.

## References

  * [1] K. J. Böröczky, Á. Sagmeister: The isodiametric problem on the sphere and in the hyperbolic space. Acta Math. Hung., 160 (2020), 13-32.
  * [2] K. J. Böröczky, Á. Sagmeister: Convex bodies of constant width in spaces of constant curvature and the extremal area of Reuleaux triangles. Studia Scientiarum Mathematicarum Hungarica, 2022.
  * [3] K. J. Böröczky, Á. Sagmeister: Stability of the isodiametric problem on the sphere and in the hyperbolic space. Advances in Applied Mathematics, 145 (2023)
  * [4] B. V. Dekster: Completeness and constant width in spherical and hyperbolic spaces. Acta math. Hung., 67.4 (1995), 289-300.
  * [5] B. V. Dekster, The Jung theorem for spherical and hyperbolic spaces. Acta Math. Hungar., 67.4 (1995), 315–331.
  * [6] J. P. Fillmore: Barbier’s theorem in the Lobachevsky plane. Proc. Amer. Math. Soc., 24 (1970), 705-709.
  * [7] Á. G. Horváth: Diameter, width and thickness in the hyperbolic plane. Journal of Geometry, 112 (2021), no. 3, Paper No. 47, 29 pp.
  * [8] J. Jerónimo-Castro, F. G. Jimenez-Lopez: A characterization of the hyperbolic disc among constant width bodies. Bull. Korean Math. Soc., 54 (2017), 2053-2063.
  * [9] M. Lassak, M. Musielak: Spherical bodies of constant width. Aequationes mathematicae, 92 (2018), 627-640.
  * [10] K. Leichtweiss: Curves of constant width in the non-Euclidean geometry. Abh. Math. Sem. Univ. Hamburg, 75 (2005), 257-284.
  * [11] H. Martini, K. J. Swanepoel: The geometry of Minkowski spaces–A survey. Part II. Expo. Math., 22 (2004), 93-144
  * [12] L. A. Santaló: Note on convex curves in the hyperbolic plane. Bull. Am. Math. Soc. 51, 405-412 (1945)
  * [13] R. Schneider, Convex bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 2014.
  * [14] P. R. Scott: Sets of constant width and inequalities. The Quarterly Journal of Mathematics, 32(3), 345-348.