# Relations Between $p$-Means of Convex Bodies and a New Suggestion for the
Definition of the Geometric Mean

René Brandenberg  and  Florian Grundbacher

(Date: December 29, 2023.)

###### Abstract.

In light of the log-Brunn-Minkowski conjecture, various attempts have been
made to define the geometric mean of convex bodies. Many of these
constructions are fairly complex and/or fail to satisfy some natural
properties one would expect of such a mean. We remedy this by providing a new
geometric mean that is both technically simple and inherits all the natural
properties expected.

To improve our understanding of potential geometric mean definitions, we first
study general $p$-means of convex bodies, with the usual definition extended
to two series ranging over all $p\in[-\infty,\infty]$. We characterize their
equality cases and obtain (in almost all instances tight) inequalities that
quantify how well these means approximate each other. As a corollary, we
establish that every Minkowski centered body is equidistant from all its
$p$-symmetrizations with respect to the Banach-Mazur distance.

Finally, we show that our geometric mean satisfies all the properties
considered in recent literature and extend this list with some properties
regarding symmetrization and asymmetry.

###### Key words and phrases:

$p$-Means, Geometric mean, Banach-Mazur distance, Covering radius, Minkowski
asymmetry

##  1\. Introduction and Notation

Since the introduction of $L_{p}$-addition by Firey in [8] and its application
to Brunn-Minkowski theory by Lutwak in [15, 16], the resulting $L_{p}$-Brunn-
Minkowski theory has been an essential field of study in convex geometry. A
general overview of the topic is presented in [24, Chapter 9]. See also [1]
and the references therein for more recent developments.

Especially in light of the log-Brunn-Minkowski conjecture posed in [2], the
question of how the geometric mean of convex bodies should be defined
(re-)emerged, both as a tool for attacking the conjecture (cf. [6]) and to
provide a richer family of constructions of convex bodies. For the latter
goal, multiple approaches have been presented in recent literature (cf. [19,
20, 22]). However, they fail to satisfy some properties one would naturally
expect of a geometric mean or are defined in fairly complicated ways such that
they become difficult to work with. The aim of this work is to support a new
geometric mean of convex bodies that fulfills both: technical simplicity and
inheriting all the natural properties expected.

To be able to define our geometric mean, we first consider two dual classes of
$p$-means of convex bodies as introduced in [8, 9], which are closely related
to the (radial) $p$-sums studied in the (dual) $L_{p}$-Brunn-Minkowski theory.
While radial $p$-sums have been considered for
$p\in[-\infty,\infty]\setminus\\{0\\}$ before (cf. [24, Chapter 9]), $p$-sums
are usually defined only for $p\in[1,\infty]$ due to their behavior in
dependence on $p$. The case $p\in[0,1)$ has only recently gained more
attention, mostly through the log-Brunn-Minkowski conjecture [5, 13, 21].

For our purposes, we are especially interested in the case $p=0$. We therefore
extend the definitions of these $p$-means to two series ranging over all
$p\in[-\infty,\infty]$ and study how they are related to each other,
generalizing several results obtained in [3].

Let us introduce some notation. We assume $n\in\mathbb{N}$ throughout. For a
number $m\in\mathbb{N}$, we write $[m]\coloneqq\\{1,...,m\\}$. The _standard
unit vectors_ in $\mathbb{R}^{n}$ are denoted by $u^{1},...,u^{n}$. For $p\geq
1$, the $n$-dimensional _unit ball_ of the _$p$ -norm_
$\left\|\cdot\right\|_{p}$ is given by $\mathbb{B}_{p}^{n}$ with _unit sphere_
$\mathbb{S}_{p}^{n}=\operatorname{bd}(\mathbb{B}_{p}^{n})$, where we omit the
dimension if it is clear from context.

For $X,Y\subset\mathbb{R}^{n}$, $z\in\mathbb{R}^{n}$,
$\Lambda\subset\mathbb{R}$, and $\mu\in\mathbb{R}$ we define
$X+Y\coloneqq\\{x+y:x\in X,y\in Y\\}$, $X+z\coloneqq z+X\coloneqq\\{z\\}+X$,
$\Lambda X\coloneqq\\{\lambda x:\lambda\in\Lambda,x\in X\\}$, $\Lambda
z\coloneqq\Lambda\\{z\\}$, and $\mu X\coloneqq\\{\mu\\}X$. We abbreviate
$(-1)X$ and $X+((-1)Y)$ by $-X$ and $X-Y$. The _boundary_ and _interior_ of
$X$ are denoted by $\operatorname{bd}(X)$ and $\operatorname{int}(X)$. We
write $\operatorname{lin}(X)$, $\operatorname{aff}(X)$,
$\operatorname{pos}(X)$, and $\operatorname{conv}(X)$ for the _linear_ ,
_affine_ , _positive_ , and _convex hull_ of $X$, respectively.

For $a\in\mathbb{R}^{n}\setminus\\{0\\}$ and $\beta\in\mathbb{R}$, the
_hyperplane_ $H_{(a,\beta)}$ is given by
$\\{x\in\mathbb{R}^{n}:a^{T}x=\beta\\}$. We denote by
$H_{(a,\beta)}^{\leq}\coloneqq\\{x\in\mathbb{R}^{n}:a^{T}x\leq\beta\\}$ and
$H_{(a,\beta)}^{\geq}\coloneqq\\{x\in\mathbb{R}^{n}:a^{T}x\geq\beta\\}$ the
_halfspaces_ bounded by $H_{(a,\beta)}$. A halfspace
$H^{\leq}\subset\mathbb{R}^{n}$ _supports_ a set $X\subset\mathbb{R}^{n}$ at
$x\in X$ if $X\subset H^{\leq}$ and $x\in\operatorname{bd}(H^{\leq})$, and a
hyperplane $H$ supports $X$ at $x$ if one of the halfspaces bounded by $H$
does.

We denote by $\mathcal{C}^{n}$ the set of _(convex) bodies_ in
$\mathbb{R}^{n}$, i.e. non-empty compact convex sets, and by
$\mathcal{C}^{n}_{0}$ the set of bodies containing the origin in their
interior. For any $C\in\mathcal{C}^{n}$, its _polar_ is defined as
$C^{\circ}\coloneqq\\{a\in\mathbb{R}^{n}\mid\forall x\in C:a^{T}x\leq 1\\}$.
The set of _extreme points_ of $C$ is given by $\operatorname{ext}(C)$. For
$x\in C$, $N_{C}(x)$ is the _cone of outer normals_ of $x$ w.r.t. $C$. We
write $\operatorname{vol}(C)$ for the _volume_ of $C$.

For convex bodies $K,L\in\mathcal{C}^{n}$, their _Hausdorff distance_ is given
by

$d_{H}(K,L)\coloneqq\inf\\{\lambda\geq 0:K\subset
L+\lambda\mathbb{B}_{2},L\subset K+\lambda\mathbb{B}_{2}\\}$

and their _Banach-Mazur distance_ by

$d_{BM}(K,L)\coloneqq\inf\\{\lambda\geq 0:K+t^{1}\subset AL\subset\lambda
K+t^{2},A\in\mathbb{R}^{n\times n}\text{
regular},t^{1},t^{2}\in\mathbb{R}^{n}\\}.$

If $K\in\mathcal{C}^{n}$ and $C\in\mathcal{C}^{n}_{0}$, the _circumradius_ of
$K$ w.r.t. $C$ is given by

$R(K,C)\coloneqq\inf\\{\lambda\geq 0:K\subset\lambda
C+c,c\in\mathbb{R}^{n}\\},$

and any $c\in\mathbb{R}^{n}$ with $K\subset R(K,C)C+c$ is a _circumcenter_ of
$K$ w.r.t. $C$. $K$ is _optimally contained_ in $C$, abbreviated by
$K\subset^{opt}C$, if $K\subset C$ and $R(K,C)=1$. The _Minkowski asymmetry_
of $K$ is defined as $s(K)\coloneqq R(K,-K)$, and any $c\in\mathbb{R}^{n}$
with $(K-c)\subset-s(K)(K-c)$ is a _Minkowski center_ of $K$. $K$ is said to
be _Minkowski centered_ if $K\subset-s(K)K$. $K$ is _$c$ -symmetric_ with
_center of symmetry_ $c\in\mathbb{R}^{n}$ if $(K-c)=-(K-c)$, and _symmetric_
if it has a center of symmetry. For some properties of these notions, see e.g.
[3].

The _$p$ -mean_ of two positive real numbers $\alpha,\beta>0$ is for
$p\in\mathbb{R}\setminus\\{0\\}$ defined as

$m_{p}(\alpha,\beta)\coloneqq\left(\frac{\alpha^{p}+\beta^{p}}{2}\right)^{\nicefrac{{1}}{{p}}}.$

Moreover, the _$(-\infty)$ -_, _$0$ -_ and _$\infty$ -means_ are given by

$m_{-\infty}(\alpha,\beta)\coloneqq\min\\{\alpha,\beta\\},~{}m_{0}(\alpha,\beta)\coloneqq\sqrt{\alpha\beta},\text{
and }m_{\infty}(\alpha,\beta)\coloneqq\max\\{\alpha,\beta\\}.$

######  Remark 1.1.

It is well-known that $m_{p}(\alpha,\beta)$ is increasing and continuous in
all three arguments. Furthermore, for $p<q$, it holds
$m_{p}(\alpha,\beta)=m_{q}(\alpha,\beta)$ if and only if $\alpha=\beta$. If
$p\neq\pm\infty$ and $\alpha\leq\alpha^{\prime}$, $\beta\leq\beta^{\prime}$,
then equality holds in $m_{p}(\alpha,\beta)\leq
m_{p}(\alpha^{\prime},\beta^{\prime})$ if and only if $\alpha=\alpha^{\prime}$
and $\beta=\beta^{\prime}$. Furthermore, it is easy to see that
$m_{p}(\alpha,\beta)=1/m_{-p}(1/\alpha,1/\beta)$ and for $\gamma>0$ also
$m_{p}(\gamma\alpha,\gamma\beta)=\gamma m_{p}(\alpha,\beta)$.

We further require some properties of the map
$\phi_{p,q}\colon(0,\infty)\to\mathbb{R},\allowbreak\alpha\mapsto\frac{m_{q}(\alpha,1)}{m_{p}(\alpha,1)}$
for $p,q\in[-\infty,\infty]$. For $\alpha,\beta>0$, we obtain as immediate
consequences of the above properties

(1)
$\phi_{p,q}\left(\frac{\alpha}{\beta}\right)=\frac{m_{q}(\alpha/\beta,1)}{m_{p}(\alpha/\beta,1)}=\frac{m_{q}(\alpha,\beta)}{m_{p}(\alpha,\beta)}=\frac{m_{q}(1,\beta/\alpha)}{m_{p}(1,\beta/\alpha)}=\phi_{p,q}\left(\frac{\beta}{\alpha}\right)$

as well as

(2)
$\phi_{-q,-p}(\alpha)=\frac{m_{-p}(\alpha,1)}{m_{-q}(\alpha,1)}=\frac{m_{-p}(1/\alpha,1)}{m_{-q}(1/\alpha,1)}=\frac{m_{q}(\alpha,1)}{m_{p}(\alpha,1)}=\phi_{p,q}(\alpha).$

Moreover, $\phi_{p,q}$ is clearly differentiable on $(1,\infty)$ as the
quotient of two differentiable maps and if $p$ and $q$ do not lie in
$\\{-\infty,0,\infty\\}$, its derivative is

$\displaystyle\phi_{p,q}^{\prime}(\alpha)$
$\displaystyle=\frac{\left(\frac{\alpha^{q}+1}{2}\right)^{\nicefrac{{1}}{{q}}-1}\cdot\frac{\alpha^{q-1}}{2}\cdot
m_{p}(\alpha,1)-\left(\frac{\alpha^{p}+1}{2}\right)^{\nicefrac{{1}}{{p}}-1}\cdot\frac{\alpha^{p-1}}{2}\cdot
m_{q}(\alpha,1)}{m_{p}(\alpha,1)^{2}}$
$\displaystyle=\frac{m_{q}(\alpha,1)\cdot
m_{p}(\alpha,1)\cdot\left(\frac{\alpha^{q-1}}{\alpha^{q}+1}-\frac{\alpha^{p-1}}{\alpha^{p}+1}\right)}{m_{p}(\alpha,1)^{2}}=\frac{\phi_{p,q}(\alpha)}{\alpha}\cdot\left(\frac{1}{1+\alpha^{-q}}-\frac{1}{1+\alpha^{-p}}\right).$

Defining $\frac{1}{1+\alpha^{-\infty}}\coloneqq 1$ and
$\frac{1}{1+\alpha^{\infty}}\coloneqq 0$ for all $\alpha>1$, it can easily be
verified that the final formula above for $\phi_{p,q}^{\prime}(a)$ is correct
for all $p,q\in[-\infty,\infty]$. Hence, $\phi_{p,q}^{\prime}(\alpha)\geq 0$
for $\alpha>1$ if and only if $p\leq q$. Together with (1), $\phi_{p,q}$ is
increasing on $[1,\infty)$ and decreasing on $(0,1]$ for $p\leq q$.

The arguably most important special cases of $p$-means are the minimum,
harmonic mean, geometric mean, arithmetic mean, and the maximum, which one
obtains for the $p$-values $-\infty$, $-1$, $0$, $1$, and $\infty$ in the
corresponding order. They are sometimes also refered to as _standard means_.

Four of these means have well-known analogues for convex bodies
$K,L\in\mathcal{C}^{n}_{0}$ [3]: the _minimum_ $K\cap L$ ($p=-\infty$), the
_harmonic mean_ $\left(\frac{K^{\circ}+L^{\circ}}{2}\right)^{\circ}$ ($p=-1$),
the _arithmetic mean_ $\frac{K+L}{2}$ ($p=1$), and the _maximum_
$\operatorname{conv}(K\cup L)$ ($p=\infty$). These sets are increasing from
left to right as first shown in [9]. Calling the second set the harmonic mean
of $K$ and $L$ is justified by interpreting $C^{\circ}$ as “$C^{-1}$” for
$C\in\mathcal{C}^{n}_{0}$ (cf. [19]).

Notably, the geometric mean is missing. As already mentioned, there have been
attemps to find a proper definition of it in the past. However, as the authors
of these papers point out themselves, none of these constructions are
completely satisfying (see Section 7 for details.)

The definition of $p$-means of convex bodies for general values of $p$
requires two defining functions of convex bodies:

######  Definition 1.2.

Let $C\in\mathcal{C}^{n}_{0}$ be a convex body. Its _support function_ is
defined as

$h_{C}\colon\mathbb{R}^{n}\to\mathbb{R},h_{C}(a)\coloneqq\sup\\{a^{T}x:x\in
C\\}\in[0,\infty).$

Its _gauge function_ is given by

$\left\|\cdot\right\|_{C}\colon\mathbb{R}^{n}\to\mathbb{R},\left\|x\right\|_{C}\coloneqq\inf\\{\lambda>0:x\in\lambda
C\\}\in[0,\infty).$

These funtions have many important properties (cf. [24, Section 1.7]):

######  Proposition 1.3.

Let $C,K\in\mathcal{C}^{n}_{0}$, $v,w\in\mathbb{R}^{n}$, $\lambda\geq 0$ and
$A\in\mathbb{R}^{n\times n}$ regular. Then:

  * (i)

$\left\|\cdot\right\|_{C}=h_{C^{\circ}}$,

  * (ii)

$h_{C}(\lambda v)=\lambda h_{C}(v)$, $h_{AC}(v)=h_{C}(A^{T}v)$, and
$h_{-C}(v)=h_{C}(-v)$,

  * (iii)

$h_{C}(v+w)\leq h_{C}(v)+h_{C}(w)$,

  * (iv)

$h_{C}$ is convex and thus continuous, and

  * (v)

$h_{\lambda C+K}=\lambda h_{C}+h_{K}$, and $h_{C}\leq h_{K}\iff C\subset K$.

Part (i) gives us results analogous to parts (ii) - (v) for
$\left\|\cdot\right\|_{C}$, so we do not state them explicitly. In particular,
$h_{C}$ and $\left\|\cdot\right\|_{C}$ characterize $C$ in that if
$K\in\mathcal{C}^{n}_{0}$ is another convex body, then its support/gauge
function coincides with that of $C$ if and only if $K=C$.

Using the above proposition, it can be shown that for any
$K,L\in\mathcal{C}^{n}_{0}$,

$\frac{K+L}{2}=\left\\{x\in\mathbb{R}^{n}:a^{T}x\leq\frac{h_{K}(a)+h_{L}(a)}{2}\text{
for all }a\in\mathbb{S}_{2}\right\\}$

and

$\left(\frac{K^{\circ}+L^{\circ}}{2}\right)^{\circ}=\operatorname{conv}\left(\left\\{\left(\frac{\left\|x\right\|_{K}+\left\|x\right\|_{L}}{2}\right)^{-1}\cdot
x:x\in\mathbb{S}_{2}\right\\}\right).$

This motivates the following generalization of $p$-means of convex bodies:

######  Definition 1.4.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p\in[-\infty,\infty]$. The _upper
$p$-mean_ of $K$ and $L$ is defined as

$\overline{M}_{p}(K,L)\coloneqq\\{x\in\mathbb{R}^{n}:a^{T}x\leq
m_{p}(h_{K}(a),h_{L}(a))\text{ for all }a\in\mathbb{S}_{2}\\}.$

Their _lower $p$-mean_ is given by

$\underline{M}_{p}(K,L)\coloneqq\operatorname{conv}\left(\left\\{(m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L}))^{-1}\cdot
x:x\in\mathbb{S}_{2}\right\\}\right).$

Theorem 3.1 below shows that the upper and lower $p$-mean of $K$ and $L$
coincide only in some trivial cases. This justifies giving these two sets
distinct names and notation.

One can also define weighted $p$-means of convex bodies by replacing the
$p$-means of numbers above with their weighted counterparts. Since our focus
lies on the symmetrization of convex bodies, we do not do so here.
Nonetheless, let us point out that all results in Sections 2 \- 4 remain valid
for general weighted $p$-means with almost identical proofs, and our methods
in Section 5 can be adjusted to obtain similar results in this case, as well.

The upper $p$-means for $p\in[1,\infty]$ were first introduced simply as
$p$-means in [8]. The lower $p$-means for $p\in[-\infty,-1]$ are what is
called dot-means in [9]. There is no formal necessity to restrict these means
to the above ranges of $p$-values. This has mostly been done since they can be
defined implicitely in these cases using the following property:

By definition, one has $h_{\overline{M}_{p}(K,L)}\leq m_{p}(h_{K},h_{L})$ and
$\left\|\cdot\right\|_{\underline{M}_{p}(K,L)}\leq
m_{-p}(\left\|\cdot\right\|_{K},\left\|\cdot\right\|_{L})$. If
$p\in[1,\infty]$, Minkowski’s inequality shows that $m_{p}(h_{K},h_{L})$ is a
sublinear function. Thus, it is the support function of a unique convex body
(cf. [24, Theorem 1.7.1]). By the above definition, this body must be
$\overline{M}_{p}(K,L)$ and therefore
$h_{\overline{M}_{p}(K,L)}=m_{p}(h_{K},h_{L})$. For $p\in[-\infty,-1]$, we can
argue analogously to obtain
$\left\|\cdot\right\|_{\underline{M}_{p}(K,L)}=m_{-p}(\left\|\cdot\right\|_{K},\left\|\cdot\right\|_{L})$.

Evidently, we have

$\displaystyle\underline{M}_{-\infty}(K,L)$ $\displaystyle=K\cap L,$
$\displaystyle\underline{M}_{-1}(K,L)$
$\displaystyle=\left(\frac{K^{\circ}+L^{\circ}}{2}\right)^{\circ},$
$\displaystyle\overline{M}_{1}(K,L)$ $\displaystyle=\frac{K+L}{2},$
$\displaystyle\overline{M}_{\infty}(K,L)$
$\displaystyle=\operatorname{conv}(K\cup L).$

Since we want the $p$-means of $K$ and $L$ to generalize these four special
cases, it is natural to define

(3) $M_{p}(K,L)\coloneqq\begin{cases}\overline{M}_{p}(K,L)&\text{if }p>0\\\
\underline{M}_{p}(K,L)&\text{if }p<0\end{cases}.$

Nonetheless, we study upper and lower $p$-means for all values
$p\in[-\infty,\infty]$ and for brevity refer to the entirety of both series
simply as $p$-means.

We will see later that $M_{p}(K,L)\subset M_{q}(K,L)$ for
$p,q\in[-\infty,\infty]\setminus\\{0\\}$, $p<q$, with equality if and only if
$K=L$, extending the behavior of $p$-means of real numbers. It is further easy
to see (and also a consequence of Theorem 5.6 below) that $M_{p}(K,L)$ is
continuous in $p\neq 0$. However, Theorem 3.1 below shows that the set
inclusion

(4) $\lim_{p\nearrow
0}M_{p}(K,L)=\underline{M}_{0}(K,L)\subset\overline{M}_{0}(K,L)=\lim_{p\searrow
0}M_{p}(K,L)$

is in general strict, so $M_{p}(K,L)$ cannot be extended to be continuous at
$p=0$ as well.

As we see in the following lemma, this definition of $p$-means has the added
advantage that $(M_{p}(K,L))^{\circ}\allowbreak=M_{-p}(K^{\circ},L^{\circ})$
for all $p\in[-\infty,\infty]\setminus\\{0\\}$. Again interpreting the polar
as the inverse, this generalizes the behavior of $p$-means of real numbers.
Analogously, the geometric mean $M_{0}(K,L)$ should satisfy the relation
$(M_{0}(K,L))^{\circ}=M_{0}(K^{\circ},L^{\circ})$. Since
$(\overline{M}_{0}(K,L))^{\circ}=\underline{M}_{0}(K^{\circ},L^{\circ})$ and
$(\underline{M}_{0}(K,L))^{\circ}=\overline{M}_{0}(K^{\circ},L^{\circ})$ do
not coincide in general, neither $M_{0}(K,L)=\overline{M}_{0}(K,L)$ nor
$M_{0}(K,L)=\underline{M}_{0}(K,L)$ are valid choices. This is one motivation
for finding a geometric mean $M_{0}(K,L)$ that commutes with polarity and
satisfies $M_{-p}(K,L)\subset M_{0}(K,L)\subset M_{p}(K,L)$ for all $p>0$,
i.e. $\underline{M}_{0}(K,L)\subset M_{0}(K,L)\subset\overline{M}_{0}(K,L)$.

##  2\. Technical Preliminaries

We begin with some basic properties of upper and lower $p$-means:

######  Lemma 2.1.

Let $K,K^{\prime},L,L^{\prime}\in\mathcal{C}^{n}_{0}$ with $K\subset
K^{\prime}$, $L\subset L^{\prime}$, $p,q\in[-\infty,\infty]$ with $p\leq q$,
and $A\in\mathbb{R}^{n\times n}$ regular. Then:

  * (i)

$(\overline{M}_{p}(K,L))^{\circ}=\underline{M}_{-p}(K^{\circ},L^{\circ})$ and
$(\underline{M}_{p}(K,L))^{\circ}=\overline{M}_{-p}(K^{\circ},L^{\circ})$,

  * (ii)

$\underline{M}_{p}(K,L)\subset\overline{M}_{p}(K,L)$,

  * (iii)

$\overline{M}_{p}(K,L)\subset\overline{M}_{q}(K^{\prime},L^{\prime})$ and
$\underline{M}_{p}(K,L)\subset\underline{M}_{q}(K^{\prime},L^{\prime})$, and

  * (iv)

$\overline{M}_{p}(AK,AL)=A\cdot\overline{M}_{p}(K,L)$ and
$\underline{M}_{p}(AK,AL)=A\cdot\underline{M}_{p}(K,L)$.

Note that the assumption that $A$ is regular is necessary since otherwise
$\operatorname{int}(AK)=\operatorname{int}(AL)=\emptyset$, in which case
$\overline{M}_{p}(AK,AL)$ and $\underline{M}_{p}(AK,AL)$ are not defined.

###### Proof.

For (i), we observe that

$\displaystyle\left(\overline{M}_{p}(K,L)\right)^{\circ}$
$\displaystyle=\left(\bigcap_{a\in\mathbb{S}_{2}}H_{(a,m_{p}(h_{K}(a),h_{L}(a)))}^{\leq}\right)^{\circ}=\operatorname{cl}\left(\operatorname{conv}\left(\bigcup_{a\in\mathbb{S}_{2}}\left(H_{(a,m_{p}(h_{K}(a),h_{L}(a)))}^{\leq}\right)^{\circ}\right)\right)$
$\displaystyle=\operatorname{cl}\left(\operatorname{conv}\left(\left\\{\frac{a}{m_{p}(h_{K}(a),h_{L}(a))}:a\in\mathbb{S}_{2}\right\\}\right)\right)$

The set that we take the closure and convex hull of in the last step is
compact by compactness of $\mathbb{S}_{2}$ and continuity of
$m_{p}(h_{K},h_{L})$. Thus, we may drop the closure in the last step. Since
$h_{K}=\left\|\cdot\right\|_{K^{\circ}}$ and
$h_{L}=\left\|\cdot\right\|_{L^{\circ}}$ by Proposition 1.3 (i), we conclude
that
$(\overline{M}_{p}(K,L))^{\circ}=\underline{M}_{-p}(K^{\circ},L^{\circ})$.
Now, (i) follows from
$(\underline{M}_{p}(K,L))^{\circ}=(\underline{M}_{-(-p)}((K^{\circ})^{\circ},(L^{\circ})^{\circ}))^{\circ}=((\overline{M}_{-p}(K^{\circ},L^{\circ}))^{\circ})^{\circ}=\overline{M}_{-p}(K^{\circ},L^{\circ})$.

For (ii), let $x,a\in\mathbb{S}_{2}$. Then $a^{T}x\leq 0$, or $a^{T}x>0$ and

$\displaystyle\frac{a^{T}x}{m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})}$
$\displaystyle=\frac{1}{m_{-p}\left(\frac{\left\|x\right\|_{K}}{a^{T}x},\frac{\left\|x\right\|_{L}}{a^{T}x}\right)}=m_{p}\left(a^{T}\frac{x}{\left\|x\right\|_{K}},a^{T}\frac{x}{\left\|x\right\|_{L}}\right)\leq
m_{p}(h_{K}(a),h_{L}(a)).$

Since $a\in\mathbb{S}_{2}$ has been chosen arbitrarily, we obtain
$x/m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})\in\overline{M}_{p}(K,L)$.
Since $x\in\mathbb{S}_{2}$ has also been chosen arbitrarily and
$\overline{M}_{p}(K,L)$ is clearly convex, (ii) is proved.

The first inclusion in (iii) is an immediate consequence of the monotonicity
of $p$-means of real numbers and Proposition 1.3 (v). The second inclusion
follows from (i).

Finally, for (iv), let $x\in\mathbb{R}^{n}$. Then $x$ lies in
$\overline{M}_{p}(AK,AL)$ if and only if $a^{T}x\leq
m_{p}(h_{AK}(a),h_{AL}(a))$ for all $a\in\mathbb{R}^{n}$. This is equivalent
to $(A^{T}a)^{T}(A^{-1}x)\leq m_{p}(h_{K}(A^{T}a),h_{L}(A^{T}a))$ for all
$a\in\mathbb{R}^{n}$, i.e. $A^{-1}x\in\overline{M}_{p}(K,L)$. Consequently,
$\overline{M}_{p}(AK,AL)=A\cdot\overline{M}_{p}(K,L)$. Since for any
$C\in\mathcal{C}^{n}_{0}$, it holds $(AC)^{\circ}=A^{-T}\cdot C^{\circ}$, we
also obtain the second equality in (iv) from (i). ∎

Characterizing under which conditions equality holds in (ii) and (iii) above
is our next goal. We first ask under which conditions the unique boundary
points of two $p$-means on a ray $[0,\infty)x$ for
$x\in\mathbb{R}^{n}\setminus\\{0\\}$ coincide, which is equivalent to the
equality of the respective gauge function values. Since two of these means are
equal if and only if these boundary points are equal for all
$x\in\mathbb{R}^{n}\setminus\\{0\\}$, we achieve our goal using the upcomming
two lemmas. Additionally, these lemmas yield by Proposition 1.3 (i) and Lemma
2.1 (i) results on the equality of support functions, which we do not state
explicitely.

We begin by comparing upper and lower $p$-means for the same finite $p$-value.
The cases for $p=\pm\infty$ are covered by Theorem 3.1 below. The containments
from right to left and the implication from (iii) to (i) and (ii) in the
following lemma are for $p=0$ also shown in [19, Proposition 15]. Note that
while the second set is clearly contained in the first one, the reverse
inclusion is non-trivial for $p<1$ since
$h_{\overline{M}_{p}(K,L)}(a)<m_{p}(h_{K}(a),h_{L}(a))$ is possible for
general $a\in\mathbb{R}^{n}$ in that case.

######  Lemma 2.2.

Let $K,L\in\mathcal{C}^{n}_{0}$, $p\in\mathbb{R}$, and
$x\in\mathbb{R}^{n}\setminus\\{0\\}$. Then

$\displaystyle
N_{\overline{M}_{p}(K,L)}\left(\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\right)$
$\displaystyle=\left\\{a\in\mathbb{R}^{n}:a^{T}\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}=m_{p}(h_{K}(a),h_{L}(a))\right\\}$
$\displaystyle=N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right).$

Moreover, the following are equivalent:

  * (i)

$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{p}(K,L)}$,

  * (ii)

$\left\|x\right\|_{\overline{M}_{p}(K,L)}=m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})$,
and

  * (iii)

$N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)\supsetneq\\{0\\}$.

###### Proof.

By Lemma 2.1 and the definition of $\underline{M}_{p}(K,L)$, we have

$\left\|x\right\|_{\overline{M}_{p}(K,L)}\leq\left\|x\right\|_{\underline{M}_{p}(K,L)}\leq
m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L}).$

This already yields the implication from (ii) to (i). The definition of
$\overline{M}_{p}(K,L)$ now implies for any $a\in\mathbb{R}^{n}$ that
$a^{T}x\leq 0<m_{p}(h_{K}(a),h_{L}(a))$, or $a^{T}x>0$ and

(5) $\frac{a^{T}x}{m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})}\leq
a^{T}\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\leq
a^{T}\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\leq
h_{\overline{M}_{p}(K,L)}(a)\leq m_{p}(h_{K}(a),h_{L}(a)).$

If $a\in N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)$, we additionally obtain

$m_{p}(h_{K}(a),h_{L}(a))=m_{p}\left(a^{T}\frac{x}{\left\|x\right\|_{K}},a^{T}\frac{x}{\left\|x\right\|_{L}}\right)=\frac{a^{T}x}{m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})},$

and therefore equality in the full chain in (5) as well. Hence, the
implication from (iii) to (ii) and the inclusions from right to left in the
lemma follow.

Next, we assume
$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{p}(K,L)}$.
By the definition of $\underline{M}_{p}(K,L)$, there exist some
$v^{1},...,v^{k}\in\mathbb{S}_{2}$ and $\lambda_{1},...,\lambda_{k}>0$ such
that $\sum_{i\in[k]}\lambda_{i}=1$ and

(6)
$y\coloneqq\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}=\sum_{i\in[k]}\lambda_{i}\frac{v^{i}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}.$

It follows directly from the definition of $\overline{M}_{p}(K,L)$ that
$\sup_{a\in\mathbb{S}_{2}}\frac{a^{T}x}{m_{p}(h_{K}(a),h_{L}(a))}=\left\|x\right\|_{\overline{M}_{p}(K,L)}$,
where the supremum is attained for some $\bar{a}\in\mathbb{S}_{2}$ by
continuity in $a$. Therefore, by the assumption
$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{p}(K,L)}$,
there exists $\bar{a}\in\mathbb{S}_{2}$ with
$\bar{a}^{T}y=m_{p}(h_{K}(\bar{a}),h_{L}(\bar{a}))$. Since for all $i\in[k]$

$y^{i}\coloneqq\frac{v^{i}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}\in\underline{M}_{p}(K,L)\subset\overline{M}_{p}(K,L)\subset
H_{(\bar{a},m_{p}(h_{K}(\bar{a}),h_{L}(\bar{a})))}^{\leq}=H_{(\bar{a},\bar{a}^{T}y)}^{\leq},$

all $y^{i}$ must, because of (6), even belong to $H_{(\bar{a},\bar{a}^{T}y)}$.
Hence,

$m_{p}(h_{K}(\bar{a}),h_{L}(\bar{a}))=\bar{a}^{T}y=\bar{a}^{T}\frac{v^{i}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}=m_{p}\left(\bar{a}^{T}\frac{v^{i}}{\left\|v^{i}\right\|_{K}},\bar{a}^{T}\frac{v^{i}}{\left\|v^{i}\right\|_{L}}\right).$

Since $p\neq\pm\infty$, we obtain
$\bar{a}^{T}\frac{v^{i}}{\left\|v^{i}\right\|_{K}}=h_{K}(\bar{a})$ and
$\bar{a}^{T}\frac{v^{i}}{\left\|v^{i}\right\|_{L}}=h_{L}(\bar{a})$. With
$\mu_{i}\coloneqq\frac{\lambda_{i}\left\|v^{i}\right\|_{K}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}>0$,
this yields

$\frac{x}{\sum_{j\in[k]}\mu_{j}\cdot\left\|x\right\|_{\underline{M}_{p}(K,L)}}=\sum_{i\in[k]}\frac{\mu_{i}}{\sum_{j\in[k]}\mu_{j}}\cdot\frac{v^{i}}{\left\|v^{i}\right\|_{K}}\in\operatorname{conv}\left(\left\\{\frac{v^{1}}{\left\|v^{1}\right\|_{K}},...,\frac{v^{k}}{\left\|v^{k}\right\|_{K}}\right\\}\right)\subset
K\cap H_{(\bar{a},h_{K}(\bar{a}))}.$

It follows
$\sum_{j\in[k]}\mu_{j}\cdot\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{K}$
and $\bar{a}\in N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)$. We can
argue analogously for $\bar{a}\in
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)$. This shows the implication
from (i) to (iii) and that the second set in the lemma is contained in the
third one, as any $a\in\mathbb{R}^{n}\setminus\\{0\\}$ with
$a^{T}\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}=m_{p}(h_{K}(a),h_{L}(a))$
can be rescaled to lie in $\mathbb{S}_{2}$ and thus be a valid choice for
$\bar{a}$.

Finally, assume $a\in
N_{\overline{M}_{p}(K,L)}\left(\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\right)$.
Then

$\left\|a\right\|_{\underline{M}_{-p}(K^{\circ},L^{\circ})}=h_{\overline{M}_{p}(K,L)}(a)=h_{\underline{M}_{p}(K,L)}(a)=\left\|a\right\|_{\overline{M}_{-p}(K^{\circ},L^{\circ})},$

which by the already proven equivalence of (i) and (ii) yields

$a^{T}\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}=h_{\overline{M}_{p}(K,L)}(a)=\left\|a\right\|_{\overline{M}_{-p}(K^{\circ},L^{\circ})}=m_{p}(\left\|a\right\|_{K^{\circ}},\left\|a\right\|_{L^{\circ}})=m_{p}(h_{K}(a),h_{L}(a)).$

Thus, the first set in the lemma is contained in the second one, which
completes the proof. ∎

Next, we compare $p$-means for different $p$-values.

######  Lemma 2.3.

Let $K,L\in\mathcal{C}^{n}_{0}$, $p,q\in[-\infty,\infty]$, $p<q$, and
$x\in\mathbb{R}^{n}\setminus\\{0\\}$.

  * (i)

$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$
if and only if
$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$.

  * (ii)

$\left\|x\right\|_{\overline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{q}(K,L)}$
if and only if
$N_{\overline{M}_{q}(K,L)}\left(\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\right)\supsetneq\\{0\\}$,
where

$\displaystyle
N_{\overline{M}_{q}(K,L)}\left(\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\right)$
$\displaystyle=\left\\{a\in\mathbb{R}^{n}:a^{T}\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}=m_{q}(h_{K}(a),h_{L}(a))\right\\}$
$\displaystyle=\left\\{a\in\mathbb{R}^{n}:a^{T}\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}=h_{K}(a)=h_{L}(a)\right\\}.$

  * (iii)

If
$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{q}(K,L)}$,
then
$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\overline{M}_{q}(K,L)}$
and

$N_{\overline{M}_{q}(K,L)}\left(\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\right)=N_{\overline{M}_{\infty}(K,L)}\left(\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\right)=N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)\supsetneq\\{0\\}.$

Let us point out that while for $q\in[-\infty,-1]$ the equality
$\left\|x\right\|_{K}=\left\|x\right\|_{L}$ always also implies
$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$,
this is not necessarily the case for $q>-1$ (cf. Figure 1). Similarly, the
first equality in (ii) is again non-trivial for $q<1$. Furthermore, the second
equality in (ii) does not imply that
$\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}$ lies in $K$ or $L$ (cf.
Figure 1).

0 Figure 1.  An example for Lemma 2.3: $K$ (black), $-K$ (dashed),
$\underline{M}_{0}(K,-K)$ (orange), $\overline{M}_{1}(K,-K)$ (blue),
$\overline{M}_{3}(K,-K)$ (red), $\overline{M}_{\infty}(K,-K)$ (dashdotted).
The common boundary points of $K$ and $-K$ are not boundary points of
$\underline{M}_{0}(K,-K)$. Furthermore, the vertices of
$\overline{M}_{1}(K,-K)$ are smooth boundary points of
$\overline{M}_{\infty}(K,-K)$. By Lemma 2.3 (ii), $\overline{M}_{p}(K,-K)$ for
$p>1$ is supported at each of these points by exactly one respective line that
also supports $K$ and $-K$. However, this does not mean that these points must
belong to $K$ or $-K$.

###### Proof.

For (i), if
$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$,
then Lemma 2.1 and the definition of $\underline{M}_{p}(K,L)$ show

$\left\|x\right\|_{\underline{M}_{q}(K,L)}\leq\left\|x\right\|_{\underline{M}_{p}(K,L)}\leq
m_{-p}(\left\|x\right\|_{K},\left\|x\right\|_{L})=\left\|x\right\|_{\underline{M}_{q}(K,L)},$

which already yields
$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$.
Conversely, if we assume
$\left\|x\right\|_{\underline{M}_{p}(K,L)}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$,
then there exist $v^{1},...,v^{k}\in\mathbb{S}_{2}$ and
$\lambda_{1},...,\lambda_{k}>0$ such that $\sum_{i\in[k]}\lambda_{i}=1$ and

$y\coloneqq\frac{x}{\left\|x\right\|_{\underline{M}_{q}(K,L)}}=\sum_{i\in[k]}\lambda_{i}\frac{v^{i}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}.$

Since $y\in\operatorname{bd}(\underline{M}_{q}(K,L))$, there exists a
hyperplane $H$ that supports $\underline{M}_{q}(K,L)$ at $y$.
$\underline{M}_{p}(K,L)$ is a subset of $\underline{M}_{q}(K,L)$ by Lemma 2.1,
so all $v^{i}/m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})$ must
also lie in $H$, which shows that they are boundary points of
$\underline{M}_{q}(K,L)$ as well. Thus, for all $i\in[k]$ we obtain

$m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})=\left\|v^{i}\right\|_{\underline{M}_{q}(K,L)}\leq
m_{-q}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L}),$

which by $-q<-p$ implies $\left\|v^{i}\right\|_{K}=\left\|v^{i}\right\|_{L}$.
Hence, all

$y^{i}\coloneqq\frac{v^{i}}{m_{-p}(\left\|v^{i}\right\|_{K},\left\|v^{i}\right\|_{L})}=\frac{v^{i}}{\left\|v^{i}\right\|_{K}}=\frac{v^{i}}{\left\|v^{i}\right\|_{L}}$

lie in $K\cap L$, so $y=\sum_{i\in[k]}\lambda_{i}y^{i}$ also lies in $K\cap
L$. This shows

$\left\|x\right\|_{\underline{M}_{q}(K,L)}\leq
m_{-q}(\left\|x\right\|_{K},\left\|x\right\|_{L})\leq\max\\{\left\|x\right\|_{K},\left\|x\right\|_{L}\\}\leq\left\|x\right\|_{\underline{M}_{q}(K,L)}.$

Thus,
$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\underline{M}_{q}(K,L)}$
by $-q<-p\leq\infty$.

The equivalence in (ii) and the set inclusions from right to left are clear,
so it suffices to show that if $x\in\operatorname{bd}(\overline{M}_{p}(K,L))$,
then any $a\in N_{\overline{M}_{q}(K,L)}(x)$ satisfies
$a^{T}x=h_{K}(a)=h_{L}(a)$. Since

$\left\|a\right\|_{\underline{M}_{-q}(K^{\circ},L^{\circ})}=h_{\overline{M}_{q}(K,L)}(a)=h_{\overline{M}_{p}(K,L)}(a)=\left\|a\right\|_{\underline{M}_{-p}(K^{\circ},L^{\circ})},$

(i) implies
$\left\|a\right\|_{K^{\circ}}=\left\|a\right\|_{L^{\circ}}=\left\|a\right\|_{\underline{M}_{-p}(K^{\circ},L^{\circ})}$.
We conclude $h_{K}(a)=h_{L}(a)=h_{\overline{M}_{p}(K,L)}(a)=a^{T}x$.

For (iii), we observe that Lemma 2.1 implies

$\left\|x\right\|_{\overline{M}_{q}(K,L)}\leq\left\\{\begin{array}[]{c}\left\|x\right\|_{\underline{M}_{q}(K,L)}\\\
\left\|x\right\|_{\overline{M}_{p}(K,L)}\end{array}\right\\}\leq\left\|x\right\|_{\underline{M}_{p}(K,L)}.$

By assumption, equality holds from left to right. We may therefore apply (i)
to obtain

$\left\|x\right\|_{K}=\left\|x\right\|_{L}=\left\|x\right\|_{\underline{M}_{q}(K,L)}=\left\|x\right\|_{\overline{M}_{p}(K,L)}=\left\|x\right\|_{\overline{M}_{q}(K,L)}.$

Moreover, (ii) yields

$\displaystyle\\{0\\}$ $\displaystyle\subsetneq
N_{\overline{M}_{q}(K,L)}\left(\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\right)=\left\\{a\in\mathbb{R}^{n}:a^{T}\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}=h_{K}(a)=h_{L}(a)\right\\}$
$\displaystyle=N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)\subset
N_{\overline{M}_{\infty}(K,L)}\left(\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\right)\subset
N_{\overline{M}_{q}(K,L)}\left(\frac{x}{\left\|x\right\|_{\overline{M}_{p}(K,L)}}\right).$

Thus, the claimed equality of sets follows from
$\left\|x\right\|_{\overline{M}_{p}(K,L)}=\left\|x\right\|_{\underline{M}_{p}(K,L)}$.
∎

For general $p<q$, it is not necessarily true that
$\overline{M}_{p}(K,L)\subset\underline{M}_{q}(K,L)$. This makes it both more
difficult to handle common boundary points of these sets and less interesting
to characterize them, which is why Lemma 2.3 does not cover this case.
Fortunately, it is of least interest in terms of equality of means: As
discussed, it is natural to define “the” $p$-mean of $K$ and $L$ for positive
$p$ as $\overline{M}_{p}(K,L)$, and for negative $p$ as
$\underline{M}_{p}(K,L)$.

The following corollary collects some important consequences of the above
lemmas, which are sometimes more convenient to use than the rather technical
lemmas themselves.

######  Corollary 2.4.

Let $K,L\in\mathcal{C}^{n}_{0}$, $x\in\mathbb{R}^{n}\setminus\\{0\\}$ and
$p,q\in[-\infty,\infty]$, $p<q$. Then

  * (i)

$x\in\underline{M}_{p}(K,L)\cap\operatorname{bd}(\underline{M}_{q}(K,L))$ if
and only if
$x\in\underline{M}_{-\infty}(K,L)\cap\operatorname{bd}(\underline{M}_{q}(K,L))$.

  * (ii)

$x\in\overline{M}_{p}(K,L)\cap\operatorname{bd}(\overline{M}_{q}(K,L))$ if and
only if
$x\in\overline{M}_{p}(K,L)\cap\operatorname{bd}(\overline{M}_{\infty}(K,L))$.

  * (iii)

$x\in\underline{M}_{p}(K,L)\cap\operatorname{bd}(\overline{M}_{q}(K,L))$ if
and only if
$x\in\underline{M}_{-\infty}(K,L)\cap\operatorname{bd}(\overline{M}_{\infty}(K,L))$.

For $p\neq\pm\infty$, we have in addition:

  * (iv)

$\frac{x}{\left\|x\right\|_{\underline{M}_{p}(K,L)}}\in\operatorname{bd}(\overline{M}_{p}(K,L))$
if and only if $N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)\allowbreak\supsetneq\\{0\\}$.
Independently of $K$ and $L$, this is always the case for at least one
$x\in\mathbb{R}^{n}\setminus\\{0\\}$.

###### Proof.

The implications from right to left in (i) - (iii) are clear; the converses
follow from (i) - (iii) in Lemma 2.3. The equivalence in (iv) is already
stated in Lemma 2.2. For the claim that both statements apply for some $x$,
let $R>0$ such that $K\subset RL$ and $x$ is a common boundary point of $K$
and $RL$. Then there exists some $a\in\mathbb{R}^{n}\setminus\\{0\\}$ such
that $H_{(a,h_{RL}(a))}^{\leq}$ supports $K$ and $RL$ at $x$. Thus, $a\in
N_{K}\left(\frac{x}{\left\|x\right\|_{K}}\right)\cap
N_{L}\left(\frac{x}{\left\|x\right\|_{L}}\right)$, finishing the proof. ∎

##  3\. Equality Cases Between $p$-Means

We are now ready to analyze the equality cases of $p$-means, beginning with
upper and lower $p$-means for the same $p$-value.

######  Theorem 3.1.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p\in[-\infty,\infty]$. Then the following
are equivalent:

  * (i)

$\underline{M}_{p}(K,L)=\overline{M}_{p}(K,L)$,

  * (ii)

$p=\pm\infty$, or $n=1$, or $K$ and $L$ are dilatates of each other.

###### Proof.

We first consider the cases $p=\pm\infty$. By Lemma 2.1 (i) and (ii), we only
need to verify the inclusion
$\overline{M}_{-\infty}(K,L)\subset\underline{M}_{-\infty}(K,L)=K\cap L$. To
this end, let $x\in\overline{M}_{-\infty}(K,L)$ and $a\in\mathbb{S}_{2}$. Then
$a^{T}x\leq m_{-\infty}(h_{K}(a),h_{L}(a))=\min\\{h_{K}(a),h_{L}(a)\\}$. Since
$a$ has been chosen arbitrarily, we obtain $x\in K\cap L$.

From now on, we assume $p\neq\pm\infty$. By Corollary 2.4 (iv), (i) is
equivalent to the fact that for all $x\in\mathbb{S}_{2}$, $K$ and $L$ can be
supported by parallel hyperplanes at $\frac{x}{\left\|x\right\|_{K}}$ and
$\frac{x}{\left\|x\right\|_{L}}$, respectively. This in turn is equivalent to
$n=1$ or $K$ and $L$ being dilatates of each other, which is shown in [11]. ∎

We turn to the equality cases for different values of $p$, extending
previously known results [8, 9].

######  Theorem 3.2.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p,q\in[-\infty,\infty]$, $p<q$. Then the
following are equivalent:

  * (i)

$\underline{M}_{p}(K,L)=\underline{M}_{q}(K,L)$,

  * (ii)

$\overline{M}_{p}(K,L)=\overline{M}_{q}(K,L)$,

  * (iii)

$\underline{M}_{p}(K,L)=\overline{M}_{q}(K,L)$, and

  * (iv)

$K=L$.

###### Proof.

(iv) clearly implies (iii), which in turn implies (i) and (ii) by Lemma 2.1
(ii) and (iii). Now we show that (ii) implies (iv):
$\overline{M}_{p}(K,L)=\overline{M}_{q}(K,L)$ yields that
$x/\left\|x\right\|_{\overline{M}_{p}(K,L)}$ is a boundary point of
$\overline{M}_{q}(K,L)$ for all $x\in\mathbb{S}_{2}$. Corollary 2.4 now
implies that $\overline{M}_{r}(K,L)=\overline{M}_{\infty}(K,L)$ for all
$r\in[p,\infty]$. In particular, $m_{r}(h_{K},h_{L})=m_{\infty}(h_{K},h_{L})$
for all sufficiently large $r\in[1,\infty]$ by $p<q\leq\infty$, which is only
possible for $h_{K}=h_{L}$, i.e. $K=L$. For the implication from (i) to (iv),
we use an analogous argument involving gauge functions. ∎

Next, we consider the monotonicity of $p$-means in $K$ and $L$. For
$p=-\infty$, it is clear that $\underline{M}_{p}(K,L)$ does not change if we
replace both $K$ and $L$ by $K\cap L$. If $K\neq L$, this properly decreases
at least one of $K$ and $L$, which shows that $\underline{M}_{-\infty}$ is not
strictly increasing in its set-arguments. Example 4.2 below shows that there
is no strict monotonicity in general for $n\geq 2$ and $p\in(-1,\infty]$
either. Using Lemma 2.1 (i), we obtain analogous results for
$\overline{M}_{p}$ with $p\in[-\infty,1)\cup\\{\infty\\}$. However, we do have
strict monotonicity in the remaining cases. This is clear for $n=1$. The
higher dimensional case should be equally well-known, but is to our knowledge
surprisingly not stated explicitely in standard literature. We include it here
for completeness.

######  Theorem 3.3.

Let $K,K^{\prime},L,L^{\prime}\in\mathcal{C}^{n}_{0}$ with $K\subset
K^{\prime}$, $L\subset L^{\prime}$, and $p\in[1,\infty)$. Then the following
are equivalent:

  * (i)

$\underline{M}_{-p}(K,L)=\underline{M}_{-p}(K^{\prime},L^{\prime})$,

  * (ii)

$\overline{M}_{p}(K,L)=\overline{M}_{p}(K^{\prime},L^{\prime})$, and

  * (iii)

$K=K^{\prime}$, $L=L^{\prime}$.

###### Proof.

(iii) is clearly sufficient for (i) and (ii). For the implication from (ii) to
(iii), we use that $h_{\overline{M}_{p}(K,L)}=m_{p}(h_{K},h_{L})$ by $p\geq 1$
and the discussion following Definition 1.4. This also applies for
$K^{\prime}$ and $L^{\prime}$. Hence
$m_{p}(h_{K},h_{L})=m_{p}(h_{K^{\prime}},h_{L^{\prime}})$, which by
$p\in[1,\infty)$ implies $h_{K}=h_{K^{\prime}}$ and $h_{L}=h_{L^{\prime}}$.
This gives (iii). For the implication from (i) to (iii), we use an analogous
argument involving gauge functions. ∎

##  4\. $p$-Means of Polytopes

The goal of this section is to show that if $K,L\in\mathcal{C}^{n}_{0}$ are
polytopes, then some of their $p$-means are guaranteed to be polytopes, too.
We obtain explicit representations of these polytopes, based on
representations of $K$ and $L$, which allows for easier computation of the
means in question.

######  Theorem 4.1.

Let $K,L\in\mathcal{C}^{n}_{0}$ be polytopes and
$p\in[-\infty,1]\cup\\{\infty\\}$. Then $\underline{M}_{-p}(K,L)$ and
$\overline{M}_{p}(K,L)$ are polytopes as well.

Besides the trivial case $p=\infty$, the interval $(0,1]$ for $p$ in the above
proposition is of particular interest, as this is the range of $p$-values
where it applies to $M_{p}(K,L)$.

###### Proof.

We only need to show that $\underline{M}_{-p}(K,L)$ is a polytope since
$K^{\circ}$ and $L^{\circ}$ are also polytopes and
$\overline{M}_{p}(K,L)=\left(\underline{M}_{-p}(K^{\circ},L^{\circ})\right)^{\circ}$.
As $\underline{M}_{-\infty}(K,L)=K\cap L$ is clearly a polytope, we may assume
$p\in[-\infty,1]$.

Let $a,b\in\mathbb{R}^{n}$ such that $F\coloneqq K\cap H_{(a,1)}$ and
$E\coloneqq L\cap H_{(b,1)}$ are facets of the respective polytopes. It is
easy to see that for $v\in\text{pos}(F)$, it holds

(7) $\left\|v\right\|_{K}=a^{T}v.$

Now let
$v,w\in(\operatorname{pos}(F)\cap\operatorname{pos}(E))\setminus\\{0\\}$ and
write
$\bar{v}\coloneqq\frac{v}{m_{p}(\left\|v\right\|_{K},\left\|v\right\|_{L})}$
and
$\bar{w}\coloneqq\frac{w}{m_{p}(\left\|w\right\|_{K},\left\|w\right\|_{L})}$.
For $\lambda\in[0,1]$, let
$x\coloneqq\lambda\bar{v}+(1-\lambda)\bar{w}\in\underline{M}_{-p}(K,L)$. We
claim that

$\frac{x}{m_{p}(\left\|x\right\|_{K},\left\|x\right\|_{L})}\in[0,x],$

which is equivalent to

(8) $1\leq
m_{p}(\left\|\lambda\bar{v}+(1-\lambda)\bar{w}\right\|_{K},\left\|\lambda\bar{v}+(1-\lambda)\bar{w}\right\|_{L}).$

By continuity in $p$, we may restrict to $p\in(-\infty,0)\cup(0,1)$. Using
(7), the right-hand side of (8) can be rewritten as

$m_{p}\left(\frac{\lambda
a^{T}v}{m_{p}(a^{T}v,b^{T}v)}+\frac{(1-\lambda)a^{T}w}{m_{p}(a^{T}w,b^{T}w)},\frac{\lambda
b^{T}v}{m_{p}(a^{T}v,b^{T}v)}+\frac{(1-\lambda)b^{T}w}{m_{p}(a^{T}w,b^{T}w)}\right).$

Defining for $\alpha,\beta>0$ the term
$\left\|(\alpha,\beta)\right\|_{p}\coloneqq(\alpha^{p}+\beta^{p})^{\nicefrac{{1}}{{p}}}=2^{\nicefrac{{1}}{{p}}}m_{p}(\alpha,\beta)$,
we may re-express the right-hand side of (8) in a more familiar way:

$\frac{\left\|\lambda\cdot\left\|(a^{T}w,b^{T}w)\right\|_{p}\cdot(a^{T}v,b^{T}v)+(1-\lambda)\cdot\left\|(a^{T}v,b^{T}v)\right\|_{p}\cdot(a^{T}w,b^{T}w)\right\|_{p}}{\left\|(a^{T}v,b^{T}v)\right\|_{p}\cdot\left\|(a^{T}w,b^{T}w)\right\|_{p}}$

Using the reverse Minkowski inequality for $\left\|\cdot\right\|_{p}$ on
$(0,\infty)^{2}$ yields (8).

The above shows that if $\mathcal{R}(F,E)$ is the set of euclidean unit
direction vectors of all extreme rays of $\text{pos}(F)\cap\text{pos}(E)$,
then

$\displaystyle\operatorname{conv}\left(\left\\{0\right\\}\cup\left\\{\frac{v}{m_{p}(\left\|v\right\|_{K},\left\|v\right\|_{L})}:v\in\mathbb{S}_{2}\cap\text{pos}(F)\cap\text{pos}(E)\right\\}\right)$
$\displaystyle=\operatorname{conv}\left(\left\\{0\right\\}\cup\left\\{\frac{v}{m_{p}(\left\|v\right\|_{K},\left\|v\right\|_{L})}:v\in\mathcal{R}(F,E)\right\\}\right).$

Any $v\in\mathbb{S}_{2}$ lies in $\text{pos}(F)\cap\text{pos}(E)$ for some
facets $F$ and $E$ of $K$ and $L$. If we write $\mathcal{F}$ for the set of
facets of $K$ and $\mathcal{E}$ for that of $L$, we obtain that
$\underline{M}_{-p}(K,L)$ is equal to

$\operatorname{conv}\left(\left\\{\frac{v}{m_{p}(\left\|v\right\|_{K},\left\|v\right\|_{L})}:v\in\bigcup_{F\in\mathcal{F},E\in\mathcal{E}}\mathcal{R}(F,E)\right\\}\right).$

$\mathcal{F}$ and $\mathcal{E}$ are finite, and each $\mathcal{R}(E,F)$ is
finite as $\text{pos}(F)\cap\text{pos}(E)$ is a polyhedral cone. Therefore,
$\underline{M}_{-p}(K,L)$ is a polytope. ∎

For $n=2$, the proof of Theorem 4.1 shows that $\underline{M}_{p}(K,L)$ has a
fairly simple structure if $K$ and $L$ are polytopes and $p\in[-1,\infty]$:
First, note that $\frac{v}{\left\|v\right\|_{2}}$ lies in
$\bigcup_{F\in\mathcal{F},E\in\mathcal{E}}\mathcal{R}(E,F)$ for any
$v\in\operatorname{ext}(K)$, even if $n\geq 3$: Simply take any
$F\in\mathcal{F}$ that contains $v$, and any $E\in\mathcal{E}$ with
$x\in\text{pos(E)}$. Then $[0,\infty)v$ is guaranteed to be an extremal ray of
$\text{pos}(F)\cap\text{pos}(E)$. The same argument can be used for any
$v\in\operatorname{ext}(L)$. Conversely, it is easy to see that for $n=2$, the
sets $\mathcal{R}(F,E)$ consist of at most two elements of
$\mathbb{S}_{2}\cap\text{pos}(\operatorname{ext}(K)\cup\operatorname{ext}(L))$.
In summary, we have that the vertices of $\underline{M}_{p}(K,L)$ are simply
rescalations of the vertices of $K$ and $L$ when $p\in[-1,\infty]$ and $n=2$.
Using
$\overline{M}_{p}(K,L)=\left(\underline{M}_{-p}(K^{\circ},L^{\circ})\right)^{\circ}$,
the facet outer normals of $\overline{M}_{p}(K,L)$ are facet outer normals of
$K$ or $L$ under the analogous conditions.

######  Example 4.2.

Let $p\in(-1,\infty]$. Then $m_{p}(2,\frac{2}{3})>m_{-1}(2,\frac{2}{3})=1$. By
continuity, there exists $\gamma\in(0,2)$ close to $2$ with
$m_{p}(\gamma,\frac{\gamma}{\gamma+1})>1$. For $n\geq 2$, we define

$K\coloneqq\operatorname{conv}(\\{u^{n},\pm u^{i},\pm\gamma
u^{i}-u^{n}:i\in[n-1]\\})\in\mathcal{C}^{n}_{0}$

(cf. Figure 2).

$0$ Figure 2.  The bodies considered in Example 4.2 for $n=2$, $p=0$, and
$\gamma=16/9$: $K$ (black), $-K$ (dashed), $K^{\prime}$ (blue),
$\underline{M}_{p}(K,-K)$ (orange).

Applying Theorem 4.1, one obtains (with some calculations) that

(9) $\underline{M}_{p}(K,-K)=\operatorname{conv}\left(\left\\{\pm u^{n},\pm
u^{i},\frac{\pm\gamma u^{i}\pm u^{n}}{m_{-p}(\left\|\gamma
u^{1}+u^{n}\right\|_{K},1)}:i\in[n-1]\right\\}\right).$

By
$\frac{\gamma}{\gamma+1}u^{1}+\frac{1}{\gamma+1}u^{n}\in[u^{1},u^{n}]\subset
K$, it holds $\left\|\gamma u^{1}+u^{n}\right\|_{K}\leq\gamma+1$.
Additionally, $K\subset H_{(u^{1}+u^{n},1)}^{\leq}$ yields $\left\|\gamma
u^{1}+u^{n}\right\|_{K}\geq(u^{1}+u^{n})^{T}(\gamma u^{1}+u^{n})=\gamma+1$.
Altogether, we can simplify

$\frac{\pm\gamma u^{i}\pm u^{n}}{m_{-p}(\left\|\gamma
u^{1}+u^{n}\right\|_{K},1)}=\frac{\pm\gamma u^{i}\pm
u^{n}}{m_{-p}(\gamma+1,1)}=\pm
m_{p}\left(\gamma,\frac{\gamma}{\gamma+1}\right)u^{i}\pm
m_{p}\left(1,\frac{1}{\gamma+1}\right)u^{n}.$

By $m_{p}(\gamma,\frac{\gamma}{\gamma+1})>1$, we obtain $\pm
u^{i}\in\operatorname{int}(\underline{M}_{p}(K,-K))$. Thus, the example shows
that the representation of $p$-means obtained from Theorem 4.1 may be
redundant, even if we choose $L=-K$.

Moreover, with

$K^{\prime}\coloneqq\operatorname{conv}\left(\left\\{u^{n},\pm\gamma
u^{i}-u^{n},\frac{\pm\gamma
u^{i}+u^{n}}{\gamma+1}:i\in[n-1]\right\\}\right)\subsetneq K$

it holds
$\underline{M}_{p}(K^{\prime},-K^{\prime})\subset\underline{M}_{p}(K,-K)$ by
Lemma 2.1 and
$\underline{M}_{p}(K,-K)\subset\underline{M}_{p}(K^{\prime},-K^{\prime})$
since all points in the right-hand set in (9) are by definition contained in
$\underline{M}_{p}(K^{\prime},-K^{\prime})$. In summary, we have
$\underline{M}_{p}(K^{\prime},-K^{\prime})=\underline{M}_{p}(K,-K)$. This
shows that $\underline{M}_{p}$ is not strictly increasing in its two set-
arguments for $p\in(-1,\infty]$ and $n\geq 2$. Considering the polars of the
bodies constructed above yields that $\overline{M}_{p}$ is also not strictly
increasing for $p\in[-\infty,1)$ and $n\geq 2$.

##  5\. Optimal Containment and Covering Radii

In Sections 2 and 3, we have already considered the containment-relation
between $p$-means in forward direction of $p$. Inspired by [3, Theorem 1.2],
we can also obtain results on the optimal containment between $p$-means. This
requires the following proposition from [4, Theorem 2.3]:

######  Proposition 5.1.

Let $K\in\mathcal{C}^{n}$ and $C\in\mathcal{C}^{n}_{0}$. Then
$K\subset^{opt}C$ holds if and only if the following conditions are satisfied:

  * (i)

$K\subset C$, and

  * (ii)

There exist $m\in\\{2,...,n+1\\}$, $x^{1},...,x^{m}\in K$, and
$a^{1},...,a^{m}\in\mathbb{R}^{n}\setminus\\{0\\}$ such that
$H_{(a^{i},1)}^{\leq}$ supports $C$ at $x^{i}$ for every $i\in[m]$ and
$0\in\operatorname{conv}(\\{a^{1},...,a^{m}\\})$.

We turn to the optimal containment between $p$-means:

######  Theorem 5.2.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p,q\in[-\infty,\infty]$, $p<q$. Then

  * (i)

$\underline{M}_{p}(K,L)\subset^{opt}\underline{M}_{q}(K,L)$ if and only if
$\underline{M}_{-\infty}(K,L)\subset^{opt}\underline{M}_{q}(K,L)$.

  * (ii)

$\overline{M}_{p}(K,L)\subset^{opt}\overline{M}_{q}(K,L)$ if and only if
$\overline{M}_{p}(K,L)\subset^{opt}\overline{M}_{\infty}(K,L)$.

  * (iii)

$\underline{M}_{p}(K,L)\subset^{opt}\overline{M}_{q}(K,L)$ if and only if
$\underline{M}_{-\infty}(K,L)\subset^{opt}\overline{M}_{\infty}(K,L)$.

  * (iv)

$\underline{M}_{p}(K,L)\subset^{opt}\overline{M}_{p}(K,L)$ if both means are
$0$-symmetric. This applies in particular if $K$ and $L$ are $0$-symmetric, or
$L=-K$.

###### Proof.

The implications from right to left in the first three cases are direct
consequences of Lemma 2.1. We turn to their converses: For (i), Lemma 2.3 (i)
shows that if $x\in\mathbb{R}^{n}$ is a common boundary point of
$\underline{M}_{p}(K,L)$ and $\underline{M}_{q}(K,L)$, then $x$ also lies in
$\underline{M}_{-\infty}(K,L)$. Hence, any halfspace that supports
$\underline{M}_{p}(K,L)$ and $\underline{M}_{q}(K,L)$ at $x$ also supports
$\underline{M}_{-\infty}(K,L)$ at $x$. Now, Proposition 5.1 implies (i).

For (ii), Lemma 2.3 (ii) shows that if $x\in\mathbb{R}^{n}$ is a common
boundary point of $\overline{M}_{p}(K,L)$ and $\overline{M}_{q}(K,L)$, then
any halfspace that supports $\overline{M}_{p}(K,L)$ and
$\overline{M}_{q}(K,L)$ at $x$ also supports $\overline{M}_{\infty}(K,L)$ at
$x$. Using Proposition 5.1 again implies (ii).

For (iii), Lemma 2.3 (iii) shows that any common boundary point
$x\in\mathbb{R}^{n}$ of $\underline{M}_{p}(K,L)$ and $\overline{M}_{q}(K,L)$
also lies in $\underline{M}_{-\infty}(K,L)$, and that any hyperplane that
supports $\overline{M}_{q}(K,L)$ at $x$ also supports
$\overline{M}_{\infty}(K,L)$ at $x$. A third time, Proposition 5.1 yields the
desired result.

Finally, if $\underline{M}_{p}(K,L)$ and $\overline{M}_{p}(K,L)$ are both
0-symmetric, then a single common boundary point is sufficient to prove the
optimality of the containment in (iv) by Proposition 5.1. Such a point is
guaranteed to exist by Corollary 2.4 (iv) if $p\neq\pm\infty$, and by Theorem
3.1 if $p=\pm\infty$. ∎

The above theorem in particular implies that
$\underline{M}_{0}(K,-K)\subset^{opt}\overline{M}_{0}(K,-K)$ for any
$K\in\mathcal{C}^{n}_{0}$. Thus, no matter how the geometric mean $G(K,-K)$ is
defined, if it fulfills the containment chain $\underline{M}_{0}(K,-K)\subset
G(K-K)\subset\overline{M}_{0}(K,-K)$, then both set-inclusions are necessarily
optimal. The same holds true for general $L$ instead of $-K$ if $K$ and $L$
are 0-symmetric (independent of the symmetry of the geometric mean).

If we want to derive estimates on how well the $p$-means approximate each
other in backward direction of $p$-values, we require the covering radii
defined below. Those can then be used to obtain additional results for the
forward direction as well. The inequalities we obtain below generalize some of
the results in [3] as we discuss in Remark 5.12 at the end of this section.

######  Definition 5.3.

Let $K,L\in\mathcal{C}^{n}$. The _covering radius_ of $K$ w.r.t. $L$ is
defined as

$R_{0}(K,L)\coloneqq\inf\\{\lambda\geq 0\mid K\subset\lambda
L\\}\in[0,\infty].$

We also define $R_{0}^{\max}(K,L)\coloneqq\max\\{R_{0}(K,L),R_{0}(L,K)\\}$. We
sometimes abbreviate these radii by $R_{0}$ and $R_{0}^{\max}$, respectively,
if $K$ and $L$ are clear from the context.

For $L\in\mathcal{C}^{n}_{0}$, it clearly holds $R_{0}(K,L)=\max_{x\in
K}\left\|x\right\|_{L}$. An easy volume argument further shows for
$K,L\in\mathcal{C}^{n}_{0}$ that $R_{0}^{\max}(K,L)\geq 1$ with equality if
and only if $K=L$.

######  Remark 5.4.

Note that the covering radius $R_{0}(K,L)$ coincides with the circumradius
$R(K,L)$ if and only if $0$ is a circumcenter. Unlike the latter, the covering
radius depends on translations of $K$ and $L$.

Furthermore, [26, Lemma 2.1] shows that there always exist translates
$K^{\prime}$ and $L^{\prime}$ of fulldimensional bodies $K$ and $L$ such that
$R_{0}(K^{\prime},L^{\prime})=R(K,L)$, $R_{0}(L^{\prime},K^{\prime})=R(L,K)$,
and either $0\in\operatorname{int}(K^{\prime}\cap L^{\prime})$ or
$0\in\operatorname{bd}(K^{\prime})\cap\operatorname{bd}(L^{\prime})$. It is
not hard to see that this yields

(10)
$\max\\{R(K,L),R(L,K)\\}=\inf\\{R_{0}^{\max}(K-x,L-y):x\in\operatorname{int}(K),y\in\operatorname{int}(L)\\},$

where the infimum is not guaranteed to be attained as a minimum in general.

If $K$ and $L$ are $0$-symmetric, or $K$ is Minkowski centered and $L=-K$,
then equality holds in (10) even without translations. In particular, in the
latter case it holds $R_{0}^{\max}(K,-K)=s(K)$.

There is a natural connection between covering radii and support functions:

######  Lemma 5.5.

Let $f,g\colon\mathbb{S}_{2}\to(0,\infty)$ be any maps. Define
$K_{f}\coloneqq\bigcap_{a\in\mathbb{S}_{2}}H_{(a,f(a))}^{\leq}\in\mathcal{C}^{n}$
and analogously $K_{g}\in\mathcal{C}^{n}$ for $g$. Then

$R_{0}(K_{f},K_{g})\leq\sup\left\\{\frac{f(a)}{g(a)}:a\in\mathbb{S}_{2}\right\\}$

with equality if $f=h_{K_{f}}|_{\mathbb{S}_{2}}$.

Note that $R_{0}(K_{f},K_{g})=\infty$ is possible since we did not require any
additional properties of $f$ and $g$ (such as continuity).

###### Proof.

Let $s$ be the supremum in the lemma. If $s=\infty$, the inequality is clear.
Otherwise, we have for any $a\in\mathbb{S}_{2}$ that
$f(a)=\frac{f(a)}{g(a)}g(a)\leq s\cdot g(a)$. Hence,

$K_{f}\subset H_{(a,f(a))}^{\leq}\subset H_{(a,s\cdot
g(a))}^{\leq}=sH_{(a,g(a))}^{\leq},$

which proves the inequality by intersecting over all $a\in\mathbb{S}_{2}$.

Now, assume $f=h_{K_{f}}|_{\mathbb{S}_{2}}$. Let $(a^{i})_{i\in\mathbb{N}}$ be
a sequence with $a^{i}\in\mathbb{S}_{2}$ for all $i$ and
$\frac{f(a^{i})}{g(a^{i})}\to s$ $(i\to\infty)$. By
$f=h_{K_{f}}|_{\mathbb{S}_{2}}$, there exists $x^{i}\in
K_{f}\in\mathcal{C}^{n}$ with $(a^{i})^{T}x^{i}\allowbreak=f(a^{i})$ for all
$i$. Thus,

$\left\|x^{i}\right\|_{K_{g}}\geq\left\|x^{i}\right\|_{H_{(a^{i},g(a^{i}))}^{\leq}}=(x^{i})^{T}\frac{a^{i}}{g(a^{i})}=\frac{f(a^{i})}{g(a^{i})}.$

Hence, for any $0\leq\lambda<s$, there exists $i$ with $x^{i}\notin\lambda
K_{g}$, so in particular $K_{f}\not\subset\lambda K_{g}$ and therefore
$R_{0}(K_{f},K_{g})\geq s$. ∎

The assertion of the above lemma is clear from Proposition 1.3 (v) if
$f=h_{K_{f}}|_{\mathbb{S}_{2}}$ and $g=h_{K_{g}}|_{\mathbb{S}_{2}}$. However,
in the context of upper $p$-means, we sometimes only have an upper bound on
the support function. The above lemma allows us to replace the support
functions by such upper bounds, assuming they can still be used to describe
the convex bodies (as is the case for upper $p$-means).

We are now ready to prove how well $p$-means approximate each other in terms
of covering radii. The combination
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$ is dealt with
seperately, as it behaves differently from the other three and requires more
work.

######  Theorem 5.6.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p,q\in[-\infty,\infty]$. Abbreviate
$R_{0}^{\max}\coloneqq R_{0}^{\max}(K,L)$. Then

  * (i)

$\displaystyle\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}=R_{0}(\overline{M}_{p}(K,L),\overline{M}_{q}(K,L))\phantom{{}\leq
1}\text{ , if $p\geq q$,}$
$\displaystyle\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}\leq
R_{0}(\overline{M}_{p}(K,L),\overline{M}_{q}(K,L))\leq 1\text{ , if $p\leq
q$.}$

Equality is attained in the lower bound of the second case if $n=1$ or $K$ and
$L$ are dilatates of each other. Moreover, for $n\geq 2$ and any $R\geq 1$,
there exists $K\in\mathcal{C}^{n}_{0}$ with $R_{0}^{\max}(K,-K)=R$ such that
equality is attained in the upper bound for $L=-K$.

  * (ii)

all assertions in (i) are also valid for
$R_{0}(\underline{M}_{p}(K,L),\underline{M}_{q}(K,L))$ and
$R_{0}(\underline{M}_{p}(K,L),\overline{M}_{q}(K,L))$.

If $K$ and $L$ are 0-symmetric, or $K$ is Minkowski centered and $L=-K$, then
all covering radii considered above are actually circumradii. Recall that the
combination $\overline{M}_{p}(K,L)$ and $\underline{M}_{q}(K,L)$ for $p\leq q$
is harder to handle when the lower mean is used to cover the upper mean as
discussed in Section 2. In contrast, (ii) states that we know precisely by
which factor the upper mean must be dilatated to cover the lower mean.

###### Proof.

We abbreviate $R_{0}(\underline{M}_{p}(K,L),\overline{M}_{q}(K,L))$ as
$R_{0}(\underline{p},\overline{q})$ and for the other combinations
analogously. We omit the bars for $\pm\infty$ since the respective means are
equal in that case by Theorem 3.1.

We start with the case $p\geq q$. By Lemma 5.5, we have

$R_{0}^{\max}=\sup\left\\{\frac{h_{K}(a)}{h_{L}(a)},\frac{h_{L}(a)}{h_{K}(a)}\colon
a\in\mathbb{S}_{2}\right\\}.$

By the same lemma, we also have

$R_{0}(\overline{p},\overline{q})\leq\sup\left\\{\frac{m_{p}(h_{K}(a),h_{L}(a))}{m_{q}(h_{K}(a),h_{L}(a))}\colon
a\in\mathbb{S}_{2}\right\\}$

with equality at least for $p\geq 1$ by the discussion following Definition
1.4. Remark 1.1 now implies

$\sup\left\\{\frac{m_{q}(h_{K}(a),h_{L}(a))}{m_{p}(h_{K}(a),h_{L}(a))}\colon
a\in\mathbb{S}_{2}\right\\}=\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}.$

Altogether, this means

(11)
$R_{0}(\overline{p},\overline{q})\leq\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}$

with equality at least for $p\geq 1$.

We observe that for any $A,B\in\mathcal{C}^{n}_{0}$, $R_{0}(A,B)$ is equal to
$R_{0}(B^{\circ},A^{\circ})$. Hence, (2) and (11) yield

(12)
$\displaystyle\begin{split}R_{0}(\underline{p},\underline{q})&=R_{0}((\underline{M}_{q}(K,L))^{\circ},(\underline{M}_{p}(K,L))^{\circ})=R_{0}(\overline{M}_{-q}(K^{\circ},L^{\circ}),\overline{M}_{-p}(K^{\circ},L^{\circ}))\\\
&\leq\frac{m_{-q}(R_{0}^{\max},1)}{m_{-p}(R_{0}^{\max},1)}=\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}.\end{split}$

Furthermore,

(13)
$R_{0}(\underline{p},\overline{q})\leq\left\\{\begin{array}[]{c}R_{0}(\overline{p},\overline{q})\\\
R_{0}(\underline{p},\underline{q})\end{array}\right\\}\leq\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}.$

Finally, the already known equality case in (11) combined with (12) implies

(14)
$\frac{m_{\infty}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}=R_{0}(\infty,\overline{q})\leq
R_{0}(\infty,\underline{p})\cdot
R_{0}(\underline{p},\overline{q})\leq\frac{m_{\infty}(R_{0}^{\max},1)}{m_{p}(R_{0}^{\max},1)}R_{0}(\underline{p},\overline{q}),$

which means $\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}\leq
R_{0}(\underline{p},\overline{q})$. Together with (13), the equality in (i)
and (ii) for $p\geq q$ follows.

For $p\leq q$, Lemma 2.1 immediately implies the upper bound in (i) and (ii).
Since $R_{0}(\underline{p},\overline{q})$ is a lower bound to both
$R_{0}(\underline{p},\underline{q})$ and $R_{0}(\overline{p},\overline{q})$,
the argument in (14) implies the lower bound in (i) and (ii).

We turn to the equality cases. If $L=\lambda K$ for some $\lambda\geq 1$, then
clearly $R_{0}^{\max}=\lambda$ and for any $r\in[-\infty,\infty]$ also
$\underline{M}_{r}(K,L)=m_{r}(R_{0}^{\max},1)\cdot K=\overline{M}_{r}(K,L)$.
This implies tightness of the lower bounds in (i) and (ii) in this case. If
$\lambda<1$, we reverse the roles of $K$ and $L$ and arrive at the same
conclusion. Equality in the lower bounds for $n=1$ is clear.

By $R_{0}(-\infty,\infty)\leq
R_{0}(\underline{p},\overline{q})\leq\min\\{R_{0}(\underline{p},\underline{q}),R_{0}(\overline{p},\overline{q})\\}$,
it suffices to construct for $n\geq 2$ and $R\geq 1$ some
$K\in\mathcal{C}^{n}_{0}$ with $R_{0}^{\max}(K,-K)=R$ such that
$R_{0}(K\cap(-K),\operatorname{conv}(K\cup(-K)))=1$ to finish the proof. To
this end, let

$K\coloneqq\mathbb{B}_{\infty}+\frac{R-1}{R+1}u^{n}=\bigcap_{i=1}^{n-1}H_{(\pm
u^{i},1)}^{\leq}\cap H_{(u^{n},\frac{2R}{R+1})}^{\leq}\cap
H_{(-u^{n},\frac{2}{R+1})}^{\leq}.$

It is easy to see that indeed $R_{0}^{\max}(K,-K)=R$. Moreover, $u^{1}\in
K\cap(-K)\cap\operatorname{bd}(\operatorname{conv}(K\cup(-K)))$ completes the
proof of (ii). ∎

######  Theorem 5.7.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $p,q\in[-\infty,\infty]$. Abbreviate
$R_{0}^{\max}\coloneqq R_{0}^{\max}(K,L)$. Then

$\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}\leq
R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))\leq\min\\{m_{p}(R_{0}^{\max},1),m_{-q}(R_{0}^{\max},1)\\}.$

Equality is attained in the lower bound if $n=1$ or $K$ and $L$ are dilatates
of each other. Moreover, for $n\geq 2$,
$(p,q)\notin(-\infty,1)\times(-1,\infty)\cup(1,\infty)\times(-\infty,-1)$ and
any $R\geq 1$, there exists $K\in\mathcal{C}^{n}_{0}$ with
$R_{0}^{\max}(K,-K)=R$ such that equality is attained in the upper bound for
$L=-K$.

Let us remark that the lower and upper bound coincide if and only if
$p=\infty$, or $q=-\infty$, or $R_{0}^{\max}=1$. In the first two cases, we
already know the exact value of
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$ due to Theorems 3.1 and
5.6. $R_{0}^{\max}=1$ on the other hand is equivalent to $K=L$, in which case
all $p$-means trivially coincide.

###### Proof.

The lower bound follows from Theorem 5.6 and

$\frac{m_{p}(R_{0}^{\max},1)}{m_{q}(R_{0}^{\max},1)}\leq
R_{0}(\underline{M}_{p}(K,L),\overline{M}_{q}(K,L))\leq
R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L)).$

The tightness of the lower bound is shown exactly as in Theorem 5.6.

For the upper bound, we use
$\underline{M}_{q}(K,L)\supset\underline{M}_{q}(1/R_{0}^{\max}\cdot
L,L)=m_{q}(1/R_{0}^{\max},1)\cdot L$, and analogously
$\underline{M}_{q}(K,L)\allowbreak\supset m_{q}(1/R_{0}^{\max},1)\cdot K$.
Thus, by the properties of $p$-means of real numbers discussed in Remark 1.1,

$\overline{M}_{p}(K,L)\subset\overline{M}_{\infty}(K,L)\subset
m_{-q}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L).$

Also $\overline{M}_{p}(K,L)\subset\overline{M}_{p}(R_{0}^{\max}\cdot
L,L)=m_{p}(R_{0}^{\max},1)\cdot L$, and analogously for $K$. Hence,

$\overline{M}_{p}(K,L)\subset
m_{p}(R_{0}^{\max},1)\cdot\underline{M}_{-\infty}(K,L)\subset
m_{p}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L).$

Lastly, we turn to the tightness of the upper bound. If $p=\infty$ or
$q=-\infty$, then we already discussed above that the lower and upper bound
coincide. If $p=-\infty$ or $q=\infty$, then the upper bound equals 1 and we
can use Theorem 3.1 together with the tightness in Theorem 5.6. For finite $p$
and $q$, we first consider the case $(p,q)\in[1,\infty)\times[-1,\infty)$. Let
$n\geq 2$, $R\geq 1$, and define
$K\coloneqq\operatorname{conv}(\\{v,-Rv\in\mathbb{R}^{n}:v_{1},...,v_{n-1}\in\\{-1,1\\},v_{n}=1\\})$
(cf. Figure 3).

0 Figure 3.  The bodies in the proof of Theorem 5.7 for $n=2$, $p=2$, $q=1$,
and $R=3$: $K$ (black), $-K$ (dashed), $\overline{M}_{p}(K,-K)$ (blue),
$\underline{M}_{q}(K,-K)$ (orange).

Clearly, $R_{0}^{\max}(K,-K)=R$. Moreover, $Ru^{1}$ lies in
$\overline{M}_{1}(K,-K)\subset\overline{M}_{p}(K,-K)$. If $F$ is a facet of
$K$ or $-K$, then there exist $i\in[n]$ and $\sigma\in\\{\pm 1\\}$ with
$\text{pos}(F)=\text{pos}(\\{v\in\operatorname{ext}(\mathbb{B}_{\infty}):v_{i}=\sigma\\})$.
Hence, by the proof of Theorem 4.1, we obtain that

(15)
$\underline{M}_{q}(K,-K)=\operatorname{conv}\left(\left\\{\frac{v}{m_{-q}(\left\|v\right\|_{K},\left\|v\right\|_{-K})}:v\in\\{-1,1\\}^{n}\right\\}\right)=m_{q}(R,1)\mathbb{B}_{\infty}.$

Altogether,

$R_{0}(\overline{M}_{p}(K,-K),\underline{M}_{q}(K,-K))\geq\left\|Ru^{1}\right\|_{\underline{M}_{q}(K,-K)}=\frac{R}{m_{q}(R,1)}\left\|u^{1}\right\|_{\infty}=m_{-q}(R,1),$

which by $-q\leq 1\leq p$ implies equality in the upper bound for this case.

If instead $(p,q)\in(-\infty,1]\times(-\infty,-1]$, we obtain the claim from
the first case applied to $K^{\circ}$ and $L^{\circ}$ since
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))=R_{0}(\overline{M}_{-q}(K^{\circ},L^{\circ}),\underline{M}_{-p}(K^{\circ},L^{\circ}))$.
∎

The above theorem does not cover the tightness of the upper bound for $n\geq
2$, $R_{0}^{\max}>1$, and either $(p,q)\in(-\infty,1)\times(-1,\infty)$ or
$(p,q)\in(1,\infty)\times(-\infty,-1)$. We can, in fact, show that the upper
bound cannot be attained in the first case at least for $n=2$ (see Lemma 5.10
below). In the second case, we present a better bound, which is based on a
different representation of upper $p$-means for $p>1$ given in [17]:

######  Theorem 5.8.

Let $K,L\in\mathcal{C}^{n}_{0}$ and $(p,q)\in(1,\infty)\times(-\infty,-1)$.
Abbreviate $R_{0}^{\max}\coloneqq R_{0}^{\max}(K,L)$. Then

(16) $R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))\leq
2^{\frac{1}{q}-\frac{1}{p}}R_{0}\left(\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}\mathbb{B}_{\frac{p}{p-1}}^{2},\mathbb{B}_{-q}^{2}\right).$

Moreover:

  * (i)

For $n\geq 2$ and any $R\geq 1$, there exists $K\in\mathcal{C}^{n}_{0}$ with
$R_{0}^{\max}(K,-K)=R$ such that equality is attained in (16) for $L=-K$.

  * (ii)

If $q\geq\frac{-p}{p-1}$, then the right-hand side in (16) is equal to
$m_{1}(R_{0}^{\max},1)=\frac{R_{0}^{\max}+1}{2}$.

We unfortunately do not have a more explicit representation of the upper bound
in the above theorem if $q<\frac{-p}{p-1}$. However, it is clearly at least
$\frac{R_{0}^{\max}+1}{2}$, and for $R_{0}^{\max}>1$ we can show that it is
strictly less than the upper bound obtained from Theorem 5.7 (see Remark 5.9).
Let us also point out that the upper bound in Theorem 5.7 equals
$\frac{R_{0}^{\max}+1}{2}$ if $p=1$ or $q=-1$, so the transition to the upper
bound in (16) is continuous.

###### Proof.

For $p>1$, it is shown in [17] that

$\overline{M}_{p}(K,L)=2^{-\frac{1}{p}}\left\\{\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y:x\in
K,y\in L,\mu\in[0,1]\right\\}.$

Since $q\in(-\infty,-1)$, the claimed inequality follows from

$\displaystyle R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$
$\displaystyle=\max_{z\in\overline{M}_{p}(K,L)}\left\|z\right\|_{\underline{M}_{q}(K,L)}$
$\displaystyle=2^{-\frac{1}{p}}\max_{\begin{subarray}{c}x\in K\\\ y\in L\\\
\mu\in[0,1]\end{subarray}}m_{-q}\left(\left\|\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y\right\|_{K},\left\|\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y\right\|_{L}\right)$
$\displaystyle\leq
2^{-\frac{1}{p}}\max_{\mu\in[0,1]}m_{-q}\left(\mu^{\frac{p-1}{p}}+(1-\mu)^{\frac{p-1}{p}}R_{0}^{\max},\mu^{\frac{p-1}{p}}R_{0}^{\max}+(1-\mu)^{\frac{p-1}{p}}\right)$
$\displaystyle=2^{\frac{1}{q}-\frac{1}{p}}\max_{\mu\in[0,1]}\left\|R_{0}^{\max}\cdot\begin{pmatrix}(1-\mu)^{\frac{p-1}{p}}\\\
\mu^{\frac{p-1}{p}}\end{pmatrix}+\begin{pmatrix}\mu^{\frac{p-1}{p}}\\\
(1-\mu)^{\frac{p-1}{p}}\end{pmatrix}\right\|_{-q}$
$\displaystyle=2^{\frac{1}{q}-\frac{1}{p}}\max_{w\in\mathbb{R}^{2},~{}\left\|w\right\|_{\frac{p}{p-1}}\leq
1}\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}w\right\|_{-q}.$

For (i), let
$K\coloneqq\operatorname{conv}(\\{v,-Rv\in\mathbb{R}^{n}:v_{1},...,v_{n-1}\in\\{-1,1\\},v_{n}=1\\})$,
which clearly satisfies $R_{0}^{\max}(K,-K)=R$. Moreover, if we choose
$x\coloneqq R\cdot(u^{1}-u^{n})\in K$ and $y\coloneqq
R\cdot(u^{1}+u^{n})\in-K$, then
$\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y\allowbreak\in\operatorname{pos}(F)\cap\operatorname{pos}(E)$
for every $\mu\in[0,1]$ with $F$ and $E$ the facets of $K$ and $-K$ that both
intersect both rays $[0,\infty)x$ and $[0,\infty)y$. Since
$\left\|x\right\|_{-K}=\left\|y\right\|_{K}=R$, we obtain
$\left\|\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y\right\|_{K}=\mu^{\frac{p-1}{p}}+(1-\mu)^{\frac{p-1}{p}}R$
and
$\left\|\mu^{\frac{p-1}{p}}x+(1-\mu)^{\frac{p-1}{p}}y\right\|_{-K}=\mu^{\frac{p-1}{p}}R+(1-\mu)^{\frac{p-1}{p}}$
for all $\mu\in[0,1]$. Hence, equality is attained in the above estimate for
the choice $L=-K$.

Finally, assume $-q\leq\frac{p}{p-1}$. Then, for all $w\in\mathbb{R}^{2}$,

$\left\|w\right\|_{-q}=2^{\frac{1}{-q}}m_{-q}(|w_{1}|,|w_{2}|)\leq
2^{\frac{1}{-q}}m_{\frac{p}{p-1}}(|w_{1}|,|w_{2}|)=2^{-\frac{1}{q}-\frac{p-1}{p}}\left\|w\right\|_{\frac{p}{p-1}},$

with equality if $|w_{1}|=|w_{2}|$. Thus, we obtain for $w\in\mathbb{R}^{2}$
with $\left\|w\right\|_{\frac{p}{p-1}}\leq 1$ that

$2^{\frac{1}{q}-\frac{1}{p}}\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}w\right\|_{-q}\leq
2^{\frac{1}{q}-\frac{1}{p}}(R_{0}^{\max}\left\|w\right\|_{-q}+\left\|w\right\|_{-q})\leq\frac{R_{0}^{\max}+1}{2},$

with equality if $w_{1}=w_{2}$ and $\left\|w\right\|_{\frac{p}{p-1}}=1$. This
proves (ii). ∎

######  Remark 5.9.

In the above setting, the maps

$[0,1]\ni\lambda\mapsto\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{-q}\text{ and
}[0,1]\ni\lambda\mapsto\left\|\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{\frac{p}{p-1}}$

are both strictly convex. We obtain for $\lambda\in[0,1]$

$\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{-q}\leq\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}\begin{pmatrix}1\\\
0\end{pmatrix}\right\|_{-q}=2^{-\frac{1}{q}}\cdot m_{-q}(R_{0}^{\max},1)$

with equality if and only if $\lambda\in\\{0,1\\}$, and

$\left\|\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{\frac{p}{p-1}}\geq\left\|\begin{pmatrix}1/2\\\
1/2\end{pmatrix}\right\|_{\frac{p}{p-1}}=2^{-\frac{1}{p}}$

with equality if and only if $\lambda=1/2$.

Altogether, the proof of Theorem 5.8 shows

$\displaystyle R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$
$\displaystyle\leq
2^{\frac{1}{q}-\frac{1}{p}}\max_{\lambda\in[0,1]}\frac{\left\|\begin{pmatrix}R_{0}^{\max}&1\\\
1&R_{0}^{\max}\end{pmatrix}\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{-q}}{\left\|\begin{pmatrix}1-\lambda\\\
\lambda\end{pmatrix}\right\|_{\frac{p}{p-1}}}$
$\displaystyle<2^{\frac{1}{q}-\frac{1}{p}}\frac{2^{-\frac{1}{q}}\cdot
m_{-q}(R_{0}^{\max},1)}{2^{-\frac{1}{p}}}=m_{-q}(R_{0}^{\max},1)$ and thus
also $\displaystyle R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$
$\displaystyle=R_{0}(\overline{M}_{-q}(K^{\circ},L^{\circ}),\underline{M}_{-p}(K^{\circ},L^{\circ}))<m_{p}(R_{0}^{\max},1).$

We turn to the case $n\geq 2$, $R_{0}^{\max}>1$ and
$(p,q)\in(-\infty,1)\times(-1,\infty)$ for the last inequality in this
section. Here, we do not know if the upper bound given by Theorem 5.7 can ever
be tight. The following lemma states that at least for $n=2$, this is never
the case.

######  Lemma 5.10.

Let $n=2$, $(p,q)\in(-\infty,1)\times(-1,\infty)$ and
$K,L\in\mathcal{C}^{2}_{0}$ with $R_{0}^{\max}\coloneqq R_{0}^{\max}(K,L)>1$.
Then

$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))<\min\\{m_{p}(R_{0}^{\max},1),m_{-q}(R_{0}^{\max},1)\\}.$

###### Proof.

In the proof of Theorem 5.7, it is shown that

(17) $\overline{M}_{p}(K,L)\subset\operatorname{conv}(K\cup L)\subset
m_{-q}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L).$

Now, assume we would have
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))=m_{-q}(R_{0}^{\max},1)$.
Then there exists a common boundary point $x$ of $\overline{M}_{p}(K,L)$ and
$m_{-q}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L)$, which by (17) must also
be a boundary point of $\operatorname{conv}(K\cup L)$. Moreover, there exists
some vector $a\in\mathbb{S}_{2}$ such that $H_{(a,a^{T}x)}^{\leq}$ supports
all three bodies at $x$. Our goal is to contradict $x\in\overline{M}_{p}(K,L)$
by finding an appropriate $\tilde{a}\in\mathbb{R}^{2}$ with
$\tilde{a}^{T}x>h_{\overline{M}_{p}(K,L)}(\tilde{a})$.

With $(\tilde{p},\tilde{q})\coloneqq(-q,-p)\in(-\infty,1)\times(-1,\infty)$,
$R_{0}(K^{\circ},L^{\circ})=R_{0}(L,K)$, and
$R_{0}(L^{\circ},K^{\circ})=R_{0}(K,L)$ the strict inequality w.r.t.
$m_{-q}(R_{0}^{\max},1)$ then also implies

$\displaystyle R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$
$\displaystyle=R_{0}((\underline{M}_{q}(K,L))^{\circ},(\overline{M}_{p}(K,L))^{\circ})=R_{0}(\overline{M}_{\tilde{p}}(K^{\circ},L^{\circ}),\underline{M}_{\tilde{q}}(K^{\circ},L^{\circ}))$
$\displaystyle<m_{-\tilde{q}}(R_{0}^{\max},1)=m_{p}(R_{0}^{\max},1).$

Since $p<1$, Lemma 2.3 (ii) implies $h_{K}(a)=h_{L}(a)=a^{T}x$ and from
$\overline{M}_{p}(K,L)\subset\overline{M}_{1}(K,L)$, we obtain that there
exist $v\in K$ and $w\in L$ such that $x=\frac{v+w}{2}$ and
$a^{T}v=h_{K}(a)=h_{L}(a)=a^{T}w$. Now, $[v,w]\subset\operatorname{conv}(K\cup
L)\cap H_{(a,a^{T}x)}$ implies by our assumptions that
$[v,w]\subset\operatorname{bd}(m_{-q}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L))$.

For $t\in[0,1]$, define $\gamma(t)\coloneqq(1-t)v+tw$. Then
$\left[\frac{1}{R_{0}^{\max}}v,w\right]\subset L$ yields

$\frac{\gamma(t)}{(1-t)R_{0}^{\max}+t}=\frac{(1-t)R_{0}^{\max}}{(1-t)R_{0}^{\max}+t}\left(\frac{1}{R_{0}^{\max}}v\right)+\frac{t}{(1-t)R_{0}^{\max}+t}w\in
L$

and thus $\left\|\gamma(t)\right\|_{L}\leq(1-t)R_{0}^{\max}+t$. From
$[v,w]\subset\operatorname{bd}(m_{-q}(R_{0}^{\max},1)\cdot\underline{M}_{q}(K,L))$
we derive

(18)
$m_{-q}(R_{0}^{\max},1)=\left\|\gamma(t)\right\|_{\underline{M}_{q}(K,L)}\leq
m_{-q}(\left\|\gamma(t)\right\|_{K},\left\|\gamma(t)\right\|_{L})\leq
m_{-q}(\left\|\gamma(t)\right\|_{K},(1-t)R_{0}^{\max}+t).$

Now, the function

$g_{q}(t)\coloneqq\begin{cases}((R_{0}^{\max})^{-q}+1-((1-t)R_{0}^{\max}+t)^{-q})^{-\nicefrac{{1}}{{q}}}&\text{if
}q\neq 0\\\ \frac{R_{0}^{\max}}{(1-t)R_{0}^{\max}+t}&\text{if }q=0\end{cases}$

satisfies

(19) $m_{-q}(g_{q}(t),(1-t)R_{0}^{\max}+t)=m_{-q}(R_{0}^{\max},1).$

Since $m_{-q}$ is strictly increasing in its two arguments for finite $q$,
(18) implies $\left\|\gamma(t)\right\|_{K}\geq g_{q}(t)$ (cf. Figure 4).

$a$$a_{q}$$v$$\frac{v}{m_{-q}(R_{0}^{\max},1)}$$\frac{v}{R_{0}^{\max}}$$w$$\frac{w}{m_{-q}(R_{0}^{\max},1)}$$\frac{w}{R_{0}^{\max}}$$x$
Figure 4.  The situation in the proof of Lemma 5.10: None of the points on the
orange curve (which consist of $\gamma/g_{q}$ and two rays) lie in
$\operatorname{int}(K)$, so $K$ is a subset of the orange area. The two dashed
lines are parallel and support $K$ at $v$ and $L$ at $w$, respectively. The
dashdotted line and segment are also parallel. The line containing $x$
supports $\operatorname{conv}(K\cup L)$ at $x$, whereas the segment belongs to
$\operatorname{bd}(\underline{M}_{q}(K,L))$.

In particular, we obtain
$\left\|w\right\|_{K}=\left\|\gamma(1)\right\|_{K}\geq g_{q}(1)=R_{0}^{\max}$
and since $w\in L\subset R_{0}^{\max}\cdot K$, it follows
$\frac{1}{R_{0}^{\max}}w\in\operatorname{bd}(K)$. Now, $v\in K$ and
$R_{0}^{\max}>1$ shows $w\neq v$. Since
$[v,w]\subset\operatorname{bd}(\operatorname{conv}(K\cup L))$ and
$0\in\operatorname{int}(\operatorname{conv}(K\cup L))$, the line
$\operatorname{aff}(\\{v,w\\})$ cannot contain $0$, i.e. $v$ and $w$ are
linearly independent. Using Lemma 2.1 (iv), we may w.l.o.g. assume that $v$
and $w$ are orthogonal and $\left\|v\right\|_{2}=\left\|w\right\|_{2}=1$
(applying a regular linear transformation otherwise).

Next, we define $a_{q}\coloneqq
v+(1+\frac{R_{0}^{\max}-1}{(R_{0}^{\max})^{q+1}})w$ and claim for all
$t\in[0,1]$ that $a_{q}^{T}\frac{\gamma(t)}{g_{q}(t)}\leq 1$, or equivalently
$a_{q}^{T}\gamma(t)\leq g_{q}(t)$. Due to the strict monotonicity of $m_{-q}$,
it suffices to show

(20) $m_{-q}(a_{q}^{T}\gamma(t),(1-t)R_{0}^{\max}+t)\leq
m_{-q}(g_{q}(t),(1-t)R_{0}^{\max}+t).$

By (19), the orthonormality of $\\{v,w\\}$, and the definitions of $a_{q}$ and
$\gamma(t)$, this is equivalent to

$f(t)\coloneqq
m_{-q}\left(1+t\frac{R_{0}^{\max}-1}{(R_{0}^{\max})^{q+1}},R_{0}^{\max}-t(R_{0}^{\max}-1)\right)\\\
\leq m_{-q}(R_{0}^{\max},1).$

There exists some $\varepsilon>0$ such that $(-\varepsilon,1+\varepsilon)\ni
t\mapsto f(t)$ is a well-defined function. Since $-q<1$, it follows from the
reverse Minkowski inequality that $f$ is concave. Moreover, $f$ is
differentiable and it is straightforward to compute $f^{\prime}(0)=0$. Hence,
$f$ attains its maximum over $t\in[0,1]$ for $t=0$ with
$f(0)=m_{-q}(R_{0}^{\max},1)$. In summary, (20), and thus
$a_{q}^{T}\gamma(t)\leq g_{q}(t)$, is proved. Together with
$\left\|\gamma(t)\right\|_{K}\geq g_{q}(t)$, we obtain for every $y\in
K\cap\operatorname{pos}(\\{v,w\\})$

$a_{q}^{T}y\leq\sup_{t\in[0,1]}a_{q}^{T}\frac{\gamma(t)}{\left\|\gamma(t)\right\|_{K}}\leq\sup_{t\in[0,1]}a_{q}^{T}\frac{\gamma(t)}{g_{q}(t)}\leq
1,$

with equality from left to right if $y=v$.

Since $\frac{1}{R_{0}^{\max}}w\in\operatorname{bd}(K)$, $v\in K$, and
$0\in\operatorname{int}(K)$ we must have

$\displaystyle K\cap\operatorname{pos}(\\{-v,w\\})$
$\displaystyle\subset\left(v+\operatorname{pos}\left(\left\\{-v,\left(\frac{1}{R_{0}^{\max}}w-v\right)\right\\}\right)\right)\cap\operatorname{pos}(\\{-v,w\\})$
$\displaystyle=\left[0,\frac{1}{R_{0}^{\max}}w\right]+\operatorname{pos}\left(\left\\{-v,\left(\frac{1}{R_{0}^{\max}}w-v\right)\right\\}\right)$

(cf. Figure 4). From $a_{q}^{T}v=1\geq
a_{q}^{T}\frac{\gamma(1)}{g_{q}(1)}=a_{q}^{T}(\frac{1}{R_{0}^{\max}}w)$ and
$a_{q}^{T}(-v)<0$, we obtain for every $y\in
K\cap\operatorname{pos}(\\{-v,w\\})$ that $a_{q}^{T}y\leq 1$. Exchanging the
roles of $(K,v)$ and $(L,w)$ further shows

$\displaystyle L\cap\operatorname{pos}(\\{v,-w\\})$
$\displaystyle\subset\left[0,\frac{1}{R_{0}^{\max}}v\right]+\operatorname{pos}\left(\left\\{-w,\left(\frac{1}{R_{0}^{\max}}v-w\right)\right\\}\right),$
so $K\subset R_{0}^{\max}\cdot L$ yields $\displaystyle
K\cap\operatorname{pos}(\\{v,-w\\})$
$\displaystyle\subset\left[0,v\right]+\operatorname{pos}\left(\left\\{-w,\left(\frac{1}{R_{0}^{\max}}v-w\right)\right\\}\right).$

Using the orthonormality of $\\{v,w\\}$, we compute
$a_{q}^{T}(\frac{1}{R_{0}^{\max}}v-w)=\frac{1}{R_{0}^{\max}}-1-\frac{R_{0}^{\max}-1}{(R_{0}^{\max})^{q+1}}<0$
and $a_{q}^{T}(-w)<0$, which shows for all $y\in
K\cap\operatorname{pos}(\\{v,-w\\})$ that $a_{q}^{T}y\leq a_{q}^{T}v=1$.
Finally, every $y\in K\cap\operatorname{pos}(\\{-v,-w\\})$ clearly satisfies
$a_{q}^{T}y\leq 0$. Altogether, $H_{(a_{q},1)}^{\leq}$ supports $K$ at $v$
(cf. Figure 4).

Next, we argue that $H_{(a_{q},a_{q}^{T}w)}^{\leq}$ supports $L$ at $w$. Since
$L\subset\operatorname{conv}(K\cup L)$, we have

$L\cap\operatorname{pos}(\\{v,w\\})\subset\operatorname{conv}(K\cup
L)\cap\operatorname{pos}(\\{v,w\\})=\operatorname{conv}(\\{v,w,0\\}).$

Among the three points in the convex hull, $w$ uniquely has the largest inner
product with $a_{q}$ by $R_{0}^{\max}>1$. The intersections of $L$ with
$\operatorname{pos}(\\{v,-w\\})$, $\operatorname{pos}(\\{-v,w\\})$, and
$\operatorname{pos}(\\{-v,-w\\})$ are handled analogously as for $K$ (cf.
Figure 4). Altogether, it follows
$h_{K}(a_{q})=a_{q}^{T}v<a_{q}^{T}w=h_{L}(a_{q})$. In particular, $p<1$
implies

$h_{\overline{M}_{p}(K,L)}(a_{q})\leq
m_{p}(h_{K}(a_{q}),h_{L}(a_{q}))<m_{1}(h_{K}(a_{q}),h_{L}(a_{q}))=a_{q}^{T}\left(\frac{v+w}{2}\right)=a_{q}^{T}x,$

contradicting $x\in\overline{M}_{p}(K,L)$. ∎

Let us point out that almost all steps in the above proof apply for any
dimension $n\geq 2$. The only time the assumption $n=2$ is needed is when we
conclude from $\max_{y\in
K\cap\operatorname{lin}(\\{v,w\\})}a_{q}^{T}y=a_{q}^{T}v$ that
$H_{(a_{q},a_{q}^{T}v)}^{\leq}$ contains $K$ (and for the analogous statement
with $(L,w)$ instead of $(K,v)$).

######  Remark 5.11.

For $n\geq 2$ and $R\geq 1$, let
$K\coloneqq\operatorname{conv}(\\{v,-Rv\in\mathbb{R}^{n}:v_{1},...,v_{n-1}\in\\{-1,1\\},v_{n}=1\\})$
be the body considered in the proofs of tightness for Theorems 5.7 and 5.8.
There, it is shown that
$R_{0}(\overline{M}_{p}(K,-K),\underline{M}_{q}(K,-K))$ reaches the upper
bound in the respective theorem if $(p,q)\in(1,\infty)\times\mathbb{R}$, which
can be extended to $(p,q)\in[1,\infty]\times[-\infty,\infty]$ by continuity.
This can be further extended to $(p,q)\in[-\infty,1]\times[-\infty,-1]$ by the
following argument.

A somewhat lengthy, but not difficult computation involving Theorem 4.1 shows
for $p\in[-\infty,1]$ that all facets of the polytope $\overline{M}_{p}(K,-K)$
are parallel to appropriately chosen facets of $K$ or $-K$. From this, we
obtain

$\overline{M}_{p}(K,-K)=H_{(\pm
u^{n},m_{p}(R,1))}^{\leq}\cap\bigcap_{i\in[n-1]}H_{(\pm\frac{R+1}{2R}u^{i}\pm\frac{R-1}{2R}u^{n},m_{p}(R,1))}^{\leq},$

where all signs are taken independently. Using this and
$\left\|\cdot\right\|_{\underline{M}_{q}(K,-K)}=m_{-q}(\left\|\cdot\right\|_{K},\left\|\cdot\right\|_{-K})$
for $q\in[-\infty,-1]$, it can be shown that $\overline{M}_{p}(K,-K)$ and
$m_{p}(R,1)\cdot\underline{M}_{q}(K,-K)$ have
$\frac{Rm_{p}(R,1)}{m_{1}(R,1)}u^{1}$ as a common boundary point for the given
ranges of $p$ and $q$. In particular, the results in this section show that
this construction yields tightness in the upper bound on
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$ for $n\geq 2$ and all
triples
$(p,q,R_{0}^{\max})\notin[-\infty,1)\times(-1,\infty]\times(1,\infty)$.

One might expect that the largest possible value of
$R_{0}(\overline{M}_{p}(K,L),\underline{M}_{q}(K,L))$ is achieved by the above
construction for this constellation as well. For $q\in[-1,\infty]$, we already
obtained in the proof of Theorem 5.7 that
$\underline{M}_{q}(K,-K)=m_{q}(R,1)\cdot\mathbb{B}_{\infty}$ (cf. (15)).
Hence, we can compute in this case

(21)
$R_{0}^{\max}(\overline{M}_{p}(K,-K),\underline{M}_{q}(K,-K))=\frac{m_{p}(R,1)\cdot
m_{-q}(R,1)}{m_{1}(R,1)}.$

For $R>1$ and
$(p,q)\in\\{-\infty\\}\times(-1,\infty)\cup(-\infty,1)\times\\{\infty\\}$,
this is strictly less than the known optimal upper bound $1$ from Theorems 3.1
and 5.6, and, if we assume the optimal upper bound to be continuous in
$(p,q)$, it is larger than the value in (21) at least for $p$ sufficiently
close to $-\infty$ or $q$ sufficiently close to $\infty$, as well.

######  Remark 5.12.

Let us compare the bounds obtained in this section to the results in [3]:
There, the circumradii between the four standard symmetrizations of Minkowski
centered convex bodies $K\in\mathcal{C}^{n}_{0}$ are considered. [3, Theorem
1.3] shows upper bounds on these radii, which can be obtained as special cases
from Theorems 5.6 and 5.7 here. However, the lower bounds specified in [3,
Theorem 1.7] are stronger than those obtained here, which heavily relies on
the restriction to $L=-K$. Moreover, [3, Theorem 1.5] also states that the
upper bound on the circumradii between the four standard symmetrizations in
forward direction of $p$-values can be improved for $n$ even and sufficiently
large values of the Minkowski asymmetry. This contrasts the tightness results
obtained here, which show that even under the restriction to $L=-K$, there is
no change in the optimal upper bound for large values of $R_{0}^{\max}$ if we
allow $K$ to be positioned arbitrarily. This highlights that the optimality in
$-K\subset^{opt}s(K)K$ for a Minkowski centered $K\in\mathcal{C}^{n}_{0}$ is
essential for [3, Theorem 1.5].

In closing this section, let us point out that $R_{0}^{\max}$ is a
multiplicative metric on $\mathcal{C}^{n}_{0}$, i.e. $\log(R_{0}^{\max})$ is a
metric, which behaves somewhat similar to the Hausdorff metric. Therefore,
Theorems 5.6 to 5.8 can be seen as stability results concerning Theorem 3.2 in
the sense that if $K,L\in\mathcal{C}^{n}_{0}$ are close w.r.t. $R_{0}^{\max}$,
then so are their $p$-means. The equality in Theorem 5.6 for $p\geq q$ even
allows conclusions in the other direction.

##  6\. Banach-Mazur Distance

Theorem 5.6 lets us estimate the Banach-Mazur distance between $K$ (or $L$)
and the $p$-means of $K$ and $L$. The special case for $p$-symmetrizations,
i.e. when $L=-K$, is of particular interest. It extends the rather well-known
result that for any $K\in\mathcal{C}^{n}_{0}$, $\frac{K-K}{2}$ minimizes the
Banach-Mazur distance to $K$ among all symmetric convex bodies. A
generalization of this property to the four standard symmetrizations already
appears in [3, Remark 4.6].

######  Corollary 6.1.

Let $p\in[-\infty,\infty]$ and $K,L,C\in\mathcal{C}^{n}_{0}$ such that
$\underline{M}_{p}(K,L)\subset C\subset\overline{M}_{p}(K,L)$. Then

$d_{BM}(K,C)\leq R_{0}^{\max}(K,L).$

In particular, if $K$ is Minkowski centered, $L=-K$ and $C$ is symmetric, then

$d_{BM}(K,C)=s(K).$

###### Proof.

We have by Theorem 5.6

$\displaystyle K\subset\operatorname{conv}(K\cup L)$
$\displaystyle\subset\frac{R_{0}^{\max}(K,L)}{m_{p}(R_{0}^{\max}(K,L),1)}\underline{M}_{p}(K,L)\subset\frac{R_{0}^{\max}(K,L)}{m_{p}(R_{0}^{\max}(K,L),1)}C$
$\displaystyle\subset\frac{R_{0}^{\max}(K,L)}{m_{p}(R_{0}^{\max}(K,L),1)}\overline{M}_{p}(K,L)\subset
R_{0}^{\max}(K,L)\cdot(K\cap L)\subset R_{0}^{\max}(K,L)\cdot K.$

This immediately implies the upper bound.

If $K$ is Minkowski centered and $L=-K$, then $R_{0}^{\max}(K,L)=s(K)$.
Moreover, it is well-known that $d_{BM}(K,C)\geq s(K)$ if $C$ is symmetric
(see e.g. [10]). Together with the upper bound, the corollary is proved. ∎

If $K$ is not 0-symmetric, then
$\underline{M}_{p}(K,-K)\neq\overline{M}_{p}(K,-K)$ for $n\geq 2$ and
$p\in(-\infty,\infty)$ by Theorem 3.1. Thus, the relaxation to a set $C$
between these two symmetrizations in general applies to more sets than just
the two means. In particular, if we define the geometric mean of $K$ and $-K$
as any symmetric body $G\in\mathcal{C}^{n}_{0}$ with
$\underline{M}_{0}(K,-K)\subset G\subset\overline{M}_{0}(K,-K)$, then
$d_{BM}(K,G)=s(K)$ as well.

We also see that any non-symmetric (Minkowski centered) convex body is
equidistant w.r.t. the Banach-Mazur distance from an infinitely large class of
affinely non-isomorphic symmetric bodies. This seems to be of particular
interest in the light of [12], where it is shown that for $n=2$ and
$s\in[7/4,2]$, there exists a Minkowski centered body
$K\in\mathcal{C}^{2}_{0}$ with $s(K)=s$ that is at distance $s(K)$ from all
symmetric bodies w.r.t. the Banach-Mazur distance.

##  7\. The Geometric Mean

One important motivation behind trying to find an appropriate definition of
the geometric mean of convex bodies lies in the log-Brunn-Minkowski
conjecture. With our notation here, it states (in the case of equal weights,
which would imply the general case) that

(22) $\forall K,L\in\mathcal{C}^{n}_{0}\text{
0-symmetric}\colon\operatorname{vol}(\overline{M}_{0}(K,L))\geq\sqrt{\operatorname{vol}(K)\cdot\operatorname{vol}(L)}.$

One theoretically possible approach to prove this conjecture is to try to
define a geometric mean
$G\colon\mathcal{C}^{n}_{0}\times\mathcal{C}^{n}_{0}\to\mathcal{C}^{n}_{0}$
that satisfies for $K,L\in\mathcal{C}^{n}_{0}$

(23) $G(K,L)\subset\overline{M}_{0}(K,L)$

and

(24) $\text{$K,L$
0-symmetric}\implies\operatorname{vol}(G(K,L))\geq\sqrt{\operatorname{vol}(K)\cdot\operatorname{vol}(L)}.$

Such an approach is, for example, used in [6] to re-establish the validity of
(22) for unconditional bodies and bodies with $n$ symmetries.

The goal that we focus on is the extension of axiomatic descriptions of the
geometric mean for real numbers to more general structures. In [14], the cases
of matrices and metric spaces are considered among others. There have also
been multiple attempts to find a satisfying definition of the geometric mean
of convex bodies in the past (cf. [19, 20, 22]). Unfortunately, as the authors
of these papers note themselves, most of these constructions either fail to
satisfy some natural properties one would expect from such a geometric mean,
or only make sense for 0-symmetric bodies. There exists one exception: The
geometric mean considered by Rotem in [22] can be shown to satisfy almost all
properties considered in this section, except possibly (P8) below. However,
this construction is defined implicitely and involves rather technical tools
such that it becomes difficult to work with.

The aim of this section is to present and analyze a more natural and
technically rather simple construction that fulfills all those proposed
properties. Our suggestion is based on the idea of an iterative building of
means, which has been pursued by Milman and Rotem in [19]. It has also been
considered already in [7], and is one of the characterizations of the
geometric mean in [14]. For $p\in(0,\infty)$, $\alpha,\beta>0$ and
$i\in\mathbb{N}$, if one defines recursively

$\displaystyle m_{p}^{1}(\alpha,\beta)$ $\displaystyle\coloneqq
m_{p}(\alpha,\beta),$ $\displaystyle m_{-p}^{1}(\alpha,\beta)$
$\displaystyle\coloneqq m_{-p}(\alpha,\beta),$ $\displaystyle
m_{p}^{i+1}(\alpha,\beta)$ $\displaystyle\coloneqq
m_{p}(m_{p}^{i}(\alpha,\beta),m_{-p}^{i}(\alpha,\beta)),$ $\displaystyle
m_{-p}^{i+1}(\alpha,\beta)$ $\displaystyle\coloneqq
m_{-p}(m_{p}^{i}(\alpha,\beta),m_{-p}^{i}(\alpha,\beta)),$

it is not hard to show that $(m_{p}^{i}(\alpha,\beta))_{i\in\mathbb{N}}$ and
$(m_{-p}^{i}(\alpha,\beta))_{i\in\mathbb{N}}$ are converging sequences with

(25)
$\lim_{i\to\infty}m_{p}^{i}(\alpha,\beta)=\lim_{i\to\infty}m_{-p}^{i}(\alpha,\beta)=m_{0}(\alpha,\beta)=\sqrt{\alpha\beta},$

independently of $p$. In case of $p=1$ and replacing $\alpha$ and $\beta$ with
positive definite matrices, the analogous process still yields a common limit
that is known as the geometric mean of the initial matrizes, cf [14, Section
3].

In [19], this idea is extended to convex bodies for $p\geq 1$, mainly focusing
on $p=1$, by mixing upper $p$-means and lower $(-p)$-means. Unfortunately, as
demonstrated by Magazinov’s example in [19, Appendix], the resulting geometric
mean for $p=1$ does not posses the so called (geometric) scaling property
((P9), below).

To remedy this, we extend the construction to general $p\in\mathbb{R}$ and
show that this property is satisfied if we use $p=0$ instead. Recall that
upper and lower $p$-means of convex bodies coincide only in special cases by
Theorem 3.1. Thus, in contrast to real numbers, convergence of the sequences
for $p=0$ is non-trivial for convex bodies and it is unclear whether the
constructions for $p$ and $-p$ coincide.

The original proof by Milman and Rotem for convergence of the above process to
a common limit for convex bodies can be adapted for $p\in[0,1)$ (cf. Remark
7.3 below), but not for $p<0$. We therefore present a proof that works for
general $p\in\mathbb{R}$. It has the added advantage of proving a bound on the
convergence speed of the construction.

######  Theorem 7.1.

For $K,L\in\mathcal{C}^{n}_{0}$ and $p\in\mathbb{R}$, we define recursively
for $i\in\mathbb{N}\colon$

$\displaystyle\overline{M}_{p}^{1}(K,L)$
$\displaystyle\coloneqq\overline{M}_{p}(K,L),$
$\displaystyle\underline{M}_{-p}^{1}(K,L)$
$\displaystyle\coloneqq\underline{M}_{-p}(K,L),$
$\displaystyle\overline{M}_{p}^{i+1}(K,L)$
$\displaystyle\coloneqq\overline{M}_{p}(\overline{M}_{p}^{i}(K,L),\underline{M}_{-p}^{i}(K,L)),$
$\displaystyle\underline{M}_{-p}^{i+1}(K,L)$
$\displaystyle\coloneqq\underline{M}_{-p}(\overline{M}_{p}^{i}(K,L),\underline{M}_{-p}^{i}(K,L)).$

Then the sequences $(\overline{M}_{p}^{i}(K,L))_{i\in\mathbb{N}}$ and
$(\underline{M}_{-p}^{i}(K,L))_{i\in\mathbb{N}}$ converge to a common
Hausdorff limit $G_{p}(K,L)\in\mathcal{C}^{n}_{0}$.

###### Proof.

For ease of notation, we drop the arguments $K$ and $L$. Our goal is to prove
that $(\overline{M}_{p}^{i})_{i\in\mathbb{N}}$ and
$(\underline{M}_{-p}^{i})_{i\in\mathbb{N}}$ are Cauchy sequences with the same
limit. With the notation $\overline{M}_{p}^{0}\coloneqq K$ and
$\underline{M}_{-p}^{0}\coloneqq L$, it follows for all $i\in\mathbb{N}_{0}$
from induction on $j\in\mathbb{N}_{0}$ that

$\underline{M}_{-\infty}(\overline{M}_{p}^{i},\underline{M}_{-p}^{i})=\overline{M}_{p}^{i}\cap\underline{M}_{-p}^{i}\subset\left\\{\begin{array}[]{l}\overline{M}_{p}^{i+j}\vspace{5pt}\\\
\underline{M}_{-p}^{i+j}\end{array}\right\\}\subset\operatorname{conv}(\overline{M}_{p}^{i}\cup\underline{M}_{-p}^{i})=\overline{M}_{\infty}(\overline{M}_{p}^{i},\underline{M}_{-p}^{i}).$

Theorems 5.6 and 5.7 now imply for all $i\in\mathbb{N}$ and
$h,j\in\mathbb{N}_{0}$ that

$\displaystyle R_{0}^{\max}(\overline{M}_{p}^{i+h},\underline{M}_{-p}^{i+j})$
$\displaystyle\leq
R_{0}(\overline{M}_{\infty}(\overline{M}_{p}^{i},\underline{M}_{-p}^{i}),\underline{M}_{-\infty}(\overline{M}_{p}^{i},\underline{M}_{-p}^{i}))=R_{0}^{\max}(\overline{M}_{p}^{i},\underline{M}_{-p}^{i})$
$\displaystyle=R_{0}^{\max}(\overline{M}_{p}(\overline{M}_{p}^{i-1},\underline{M}_{-p}^{i-1}),\underline{M}_{-p}(\overline{M}_{p}^{i-1},\underline{M}_{-p}^{i-1}))\leq
m_{|p|}(R_{0}^{\max}(\overline{M}_{p}^{i-1},\underline{M}_{-p}^{i-1}),1).$

Induction on $i\in\mathbb{N}$ thus shows

$\displaystyle
R_{0}^{\max}(\overline{M}_{p}^{i+h},\underline{M}_{-p}^{i+j})\leq R_{p}(i)$
$\displaystyle\coloneqq\underbrace{m_{|p|}(m_{|p|}(...m_{|p|}}_{\text{$i$
times}}(R_{0}^{\max}(K,L),1)...,1),1)$ $\displaystyle{\hskip
2.0pt}=\begin{cases}\left(1+\frac{R_{0}^{\max}(K,L)^{|p|}-1}{2^{i}}\right)^{\nicefrac{{1}}{{|p|}}}&\text{if
}p\neq 0\\\ R_{0}^{\max}(K,L)^{\nicefrac{{1}}{{2^{i}}}}&\text{if
}p=0\end{cases}.$

The same holds true for the pairs $\overline{M}_{p}^{i+h}(K,L)$,
$\overline{M}_{p}^{i+j}(K,L)$ and $\underline{M}_{-p}^{i+h}(K,L)$,
$\underline{M}_{-p}^{i+j}(K,L)$. Since for any
$K^{\prime},L^{\prime}\in\mathcal{C}^{n}_{0}$ and $R^{\prime}>0$ with
$K^{\prime},L^{\prime}\subset R^{\prime}\mathbb{B}_{2}$,

$\displaystyle d_{H}(K^{\prime},L^{\prime})$
$\displaystyle\leq\sup_{x\in\mathbb{S}_{2}}\left\|\frac{x}{\left\|x\right\|_{K^{\prime}}}-\frac{x}{\left\|x\right\|_{L^{\prime}}}\right\|_{2}$
$\displaystyle\leq\sup_{x\in\mathbb{S}_{2}}\max\left\\{\frac{1}{\left\|x\right\|_{K^{\prime}}}\left(1-\frac{1}{R_{0}(K^{\prime},L^{\prime})}\right),\frac{1}{\left\|x\right\|_{L^{\prime}}}\left(1-\frac{1}{R_{0}(L^{\prime},K^{\prime})}\right)\right\\}$
$\displaystyle\leq
R^{\prime}\left(1-\frac{1}{R_{0}^{\max}(K^{\prime},L^{\prime})}\right)\leq
R^{\prime}(R_{0}^{\max}(K^{\prime},L^{\prime})-1),$

we obtain for all $i\in\mathbb{N}$ and $h,j\in\mathbb{N}_{0}$ that

(26)
$\max\\{d_{H}(\overline{M}_{p}^{i+h},\underline{M}_{-p}^{i+j}),d_{H}(\overline{M}_{p}^{i+h},\overline{M}_{p}^{i+j}),d_{H}(\underline{M}_{-p}^{i+h},\underline{M}_{-p}^{i+j})\\}\leq
R_{0}(\overline{M}_{\infty}(K,L),\mathbb{B}_{2})(R_{p}(i)-1).$

Now $R_{p}(i)\to 1$ ($i\to\infty)$ implies that
$(\overline{M}_{p}^{i})_{i\in\mathbb{N}}$ and
$(\underline{M}_{-p}^{i})_{i\in\mathbb{N}}$ are Cauchy-sequences and converge
to a common limit. Since all $\overline{M}_{p}^{i}$ contain $K\cap L$, the
joint limit contains 0 as an interior point. ∎

######  Remark 7.2.

The announced bound on the rate at which
$(\overline{M}_{p}^{i}(K,L))_{i\in\mathbb{N}}$ and
$(\underline{M}_{-p}^{i}(K,L))_{i\in\mathbb{N}}$ converge now follows from
(26):

(27)
$\max\\{d_{H}(\overline{M}_{p}^{i}(K,L),G_{p}(K,L)),d_{H}(\underline{M}_{-p}^{i}(K,L),G_{p}(K,L))\\}\leq
R_{0}(\overline{M}_{\infty}(K,L),\mathbb{B}_{2})(R_{p}(i)-1).$

We can extend $R_{p}$ to a function on $\mathbb{R}$, and see that it is
differentiable. If $R_{0}^{\max}(K,L)\neq 1$, i.e. $K\neq L$, l’Hôpital’s rule
shows

$\lim_{i\to\infty}\frac{R_{0}(\overline{M}_{\infty}(K,L),\mathbb{B}_{2})(R_{p}(i+1)-1)}{R_{0}(\overline{M}_{\infty}(K,L),\mathbb{B}_{2})(R_{p}(i)-1)}=\frac{1}{2}.$

In particular, the upper bound in (27) converges to $0$ Q-linearly with rate
$1/2$.

Furthermore, it is easy to see from the definition of $R_{p}(i)$ that it is
increasing in $|p|$. Thus, the convergence $\overline{M}_{p}^{i}(K,L)\to
G_{p}(K,L)$ ($i\to\infty$) is locally uniform in $p\in\mathbb{R}$. For fixed
$i\in\mathbb{N}$, it can additionally be shown using Theorem 5.6 that
$\overline{M}_{p}^{i}(K,L)$ is continuous in $p\in\mathbb{R}$. The Moore-
Osgood Theorem [25, pp. 139–140] therefore implies for $q\in\mathbb{R}$ that

$G_{q}(K,L)=\lim_{i\to\infty}\overline{M}_{q}^{i}(K,L)=\lim_{i\to\infty}\lim_{p\to
q}\overline{M}_{p}^{i}(K,L)=\lim_{p\to
q}\lim_{i\to\infty}\overline{M}_{p}^{i}(K,L)=\lim_{p\to q}G_{p}(K,L),$

i.e. that $G_{p}(K,L)$ is continuous in $p\in\mathbb{R}$.

######  Remark 7.3.

If $p\in[0,\infty)$, then Theorem 7.1 can be obtained adjusting the proof of
[19, Theorem 6]. It has the added advantage of proving monotonicity of the
sequences involved, which is needed later.

###### Proof of Theorem 7.1 and monotonicity of the sequences involved for
$p\in[0,\infty)$.

We again drop the arguments $K$ and $L$. Induction on $i\in\mathbb{N}$ and
Lemma 2.1 yield
$\underline{M}_{-p}^{i}\subset\underline{M}_{-p}^{i+1}\subset\overline{M}_{p}^{i+1}\subset\overline{M}_{p}^{i}$.
Thus, $(\overline{M}_{p}^{i})_{i\in\mathbb{N}}$ and
$(\underline{M}_{-p}^{i})_{i\in\mathbb{N}}$ are monotonous and bounded
sequences of convex bodies. As such, they converge to some respective
Hausdorff limits
$\overline{M}_{p}^{\infty},\underline{M}_{-p}^{\infty}\in\mathcal{C}^{n}$ with
$\underline{M}_{-p}^{1}\subset\underline{M}_{-p}^{\infty}\subset\overline{M}_{p}^{\infty}$.
In particular,
$0\in\operatorname{int}(\underline{M}_{-p}^{\infty})\cap\operatorname{int}(\overline{M}_{p}^{\infty})$
is satisfied. Moreover, for any $a\in\mathbb{R}^{n}$,

$h_{\overline{M}_{p}^{\infty}}(a)=\lim_{i\to\infty}h_{\overline{M}_{p}^{i+1}}(a)\leq\lim_{i\to\infty}m_{p}(h_{\overline{M}_{p}^{i}}(a),h_{\underline{M}_{-p}^{i}}(a))=m_{p}(h_{\overline{M}_{p}^{\infty}}(a),h_{\underline{M}_{-p}^{\infty}}(a)),$

which implies $h_{\overline{M}_{p}^{\infty}}\leq
h_{\underline{M}_{-p}^{\infty}}$ by $p\in[0,\infty)$ and consequently
$\overline{M}_{p}^{\infty}\subset\underline{M}_{-p}^{\infty}$. Thus, the
assertion is true for
$G_{p}(K,L)\coloneqq\overline{M}_{p}^{\infty}=\underline{M}_{-p}^{\infty}$. ∎

The alternative proof does not necessarily work for $p<0$ since
$\underline{M}_{-p}^{i}(K,L)\subset\overline{M}_{p}^{i}(K,L)$ and
$\overline{M}_{p}^{i}(K,L)\subset\underline{M}_{-p}^{i}(K,L)$ might both be
violated.

We will see throughout this section that all the $G_{p}$ satisfy a number of
properties that one would expect of a geometric mean, while $G_{0}$ fulfills
them all. In contrast, it is known that some of the $G_{p}$ for $p\neq 0$ fail
to satisfy (P9) (and therefore possibly also (29) below). This, and the rather
simple definition of $G_{0}$, lead us to suggest to consider it as the
geometric mean of convex bodies.

We turn to the properties of the geometric mean, first considering those
analyzed in [19, 20, 22].

######  Theorem 7.4.

Let $K,K^{\prime},L,L^{\prime}\in\mathcal{C}^{n}_{0}$ with $K\subset
K^{\prime}$, $L\subset L^{\prime}$, $A,P\in\mathbb{R}^{n\times n}$ with $A$
regular and $P$ positive definite, and $\alpha,\beta>0$. Then for all
$p\in\mathbb{R}\colon$

  1. (P1)

$G_{p}(K,K)=K$.

  2. (P2)

$G_{p}$ is symmetric in its arguments: $G_{p}(K,L)=G_{p}(L,K)$.

  3. (P3)

$G_{p}$ is increasing in both arguments: $G_{p}(K,L)\subset
G_{p}(K^{\prime},L^{\prime})$.

  4. (P4)

$G_{p}$ is continuous in both arguments w.r.t. the Hausdorff distance.

  5. (P5)

$G_{p}$ is invariant under regular linear transformation:
$G_{p}(AK,AL)=AG_{p}(K,L)$.

  6. (P6)

$G_{p}(PK,K^{\circ})=P^{\nicefrac{{1}}{{2}}}\cdot\mathbb{B}_{2}$, and in
particular $G_{p}(K,K^{\circ})=\mathbb{B}_{2}$.

  7. (P7)

$G_{p}$ commutes with polarity:
$(G_{p}(K,L))^{\circ}=G_{p}(K^{\circ},L^{\circ})$.

  8. (P8)

$\underline{M}_{0}(K,L)\subset G_{p}(K,L)\subset\overline{M}_{0}(K,L)$.

For $p=0$, we have in addition:

  1. (P8)

$\underline{M}_{0}(K,L)\subset G_{0}(K,L)\subset\overline{M}_{0}(K,L)$ with
equality on either side if and only if $n=1$ or $K$ and $L$ are dilatates of
each other.

  2. (P9)

$G_{0}$ satisfies the (geometric) scaling property: $G_{0}(\alpha K,\beta
L)=\sqrt{\alpha\beta}\cdot G_{0}(K,L)$.

In the following, we first discuss why these properties are desirable for a
geometric mean and then prove them. For (P2) \- (P8), we reuse the methods
employed in [19, 22] for $G_{1}$. For the convenience of the reader, we
restate the proof ideas (adjusted for general $p\in\mathbb{R}$), but omit the
details.

(P1) \- (P5) are often demanded for any kind of (unweighted) mean of convex
bodies.

###### Proof of (P1) \- (P5).

(P1) is an immediate consequence of
$\underline{M}_{0}(K,K)=K=\overline{M}_{0}(K,K)$ and (P8) once the latter
property is proved.

By induction, we obtain for all $i\in\mathbb{N}$ that

$\displaystyle\overline{M}_{p}^{i}(K,L)=\overline{M}_{p}^{i}(L,K)$
$\displaystyle\text{ and
}\underline{M}_{-p}^{i}(K,L)=\underline{M}_{-p}^{i}(L,K),$
$\displaystyle\overline{M}_{p}^{i}(K,L)\subset\overline{M}_{p}^{i}(K^{\prime},L^{\prime})$
$\displaystyle\text{ and
}\underline{M}_{-p}^{i}(K,L)\subset\underline{M}_{-p}^{i}(K^{\prime},L^{\prime}),$
$\displaystyle\overline{M}_{p}^{i}(AK,AL)=A\cdot\overline{M}_{p}^{i}(K,L)$
$\displaystyle\text{ and
}\underline{M}_{-p}^{i}(AK,AL)=A\cdot\underline{M}_{-p}^{i}(K,L),$

where the base cases are clear from definitions and Lemma 2.1. Using that
Hausdorff limits carry over subset relations and are continuous w.r.t. regular
linear maps, we obtain (P2), (P3), and (P5).

To see that (P4) follows from (P3) and (P5), let $(K^{i})_{i\in\mathbb{N}}$
and $(L^{i})_{i\in\mathbb{N}}$ be sequences in $\mathcal{C}^{n}_{0}$ that
converge to $K$ and $L$, respectively. For $\varepsilon\in(0,1)$, it holds
$(1-\varepsilon)K\subset K^{i}\subset(1+\varepsilon)K$ for all sufficiently
large $i$, and analogously for $L$. Thus

$(1-\varepsilon)G_{p}(K,L)=G_{p}((1-\varepsilon)K,(1-\varepsilon)L)\subset
G_{p}(K^{i},L^{i})\subset
G_{p}((1+\varepsilon)K,(1+\varepsilon)L)=(1+\varepsilon)G_{p}(K,L)$

for all sufficiently large $i$, which shows $G_{p}(K^{i},L^{i})\to G_{p}(K,L)$
($i\to\infty$). ∎

For the properties involving polars, recall that for $\alpha,\beta>0$, it
holds $(\sqrt{\alpha\beta})^{-1}=\sqrt{\alpha^{-1}\cdot\beta^{-1}}$ and
$\sqrt{\alpha\cdot\alpha^{-1}}=1$. If we interpret $C^{\circ}$ as the inverse
of a convex body $C\in\mathcal{C}^{n}_{0}$, then (P7) is the direct analog of
the first identity. The “in particular”-part of (P6) extends the second
identity with the number “1” replaced by the neutral element of polarization
$\mathbb{B}_{2}$. The first part of (P6) is a strengthening that has first
been considered in [19]. It lets us derive an explicit representation of
$G_{p}(K,L)$ if $K$ and $L$ are 0-symmetric ellipsoids (cf. [19, Proposition
10]).

###### Proof of (P6) and (P7).

By induction, we obtain for all $i\in\mathbb{N}$ that

$(\overline{M}_{p}^{i}(K,L))^{\circ}=\underline{M}_{-p}^{i}(K^{\circ},L^{\circ})\text{
and
}(\underline{M}_{-p}^{i}(K,L))^{\circ}=\overline{M}_{p}^{i}(K^{\circ},L^{\circ}),$

where the base case is clear from Lemma 2.1. Then, we use that Hausdorff
limits are continuous w.r.t. polarization on $\mathcal{C}^{n}_{0}$ to obtain
(P7).

The “in particular”-part of (P6) follows from (P2) and (P7) by

$(G_{p}(K,K^{\circ}))^{\circ}=G_{p}(K^{\circ},(K^{\circ})^{\circ})=G_{p}(K^{\circ},K)=G_{p}(K,K^{\circ}),$

which implies $G_{p}(K,K^{\circ})=\mathbb{B}_{2}$. Now, (P5) additionally
implies

$G_{p}(PK,K^{\circ})=G_{p}(P^{\nicefrac{{1}}{{2}}}P^{\nicefrac{{1}}{{2}}}K,P^{\nicefrac{{1}}{{2}}}P^{-\nicefrac{{1}}{{2}}}K^{\circ})=P^{\nicefrac{{1}}{{2}}}G_{p}(P^{\nicefrac{{1}}{{2}}}K,(P^{\nicefrac{{1}}{{2}}}K)^{\circ})=P^{\nicefrac{{1}}{{2}}}\mathbb{B}_{2}.$

∎

Since the geometric mean is the 0-mean for positive real numbers, we expect
the geometric mean of convex bodies to contain all $p$-means for $p<0$, and to
be contained in all $p$-means for $p>0$, where we use the definition of
$p$-mean in (3). Thus, (4) shows that any geometric mean should satisfy (P8),
or at the bare minimum the harmonic mean - geometric mean - arithmetic mean
inequality

$\underline{M}_{-1}(K,L)=\left(\frac{K^{\circ}+L^{\circ}}{2}\right)^{\circ}\subset
G(K,L)\subset\frac{K+L}{2}=\overline{M}_{1}(K,L).$

For $G_{0}$, we can even characterize the equality cases in (P8), which leads
us to (P8). Since we do not want a geometric mean to be biased toward
$p$-means for positive or negative values of $p$, it seems natural to demand
that a geometric mean $G$ that satisfies (P8) should only coincide with
$\underline{M}_{0}$ or $\overline{M}_{0}$ if $\underline{M}_{0}$ and
$\overline{M}_{0}$ themselves coincide. This means that $G$ should also
satisfy (P8) by Theorem 3.1.

###### Proof of (P8) and (P8).

To prove (P8), we define recursively for all $i\in\mathbb{N}_{0}\colon$

$\displaystyle\varphi_{0}$ $\displaystyle\coloneqq h_{K},$
$\displaystyle\psi_{0}$ $\displaystyle\coloneqq h_{L},$
$\displaystyle\varphi_{i+1}$ $\displaystyle\coloneqq
m_{p}(\varphi_{i},\psi_{i}),$ $\displaystyle\psi_{i+1}$
$\displaystyle\coloneqq m_{-p}(\varphi_{i},\psi_{i}).$

With the notation $\overline{M}_{p}^{0}(K,L)\coloneqq K$ and
$\underline{M}_{-p}^{0}(K,L)\coloneqq L$, it follows from induction on
$i\in\mathbb{N}_{0}$ that $h_{\overline{M}_{p}^{i}(K,L)}\leq\varphi_{i}$ and
$h_{\underline{M}_{-p}^{i}(K,L)}\leq\psi_{i}$: The case $i=0$ is clear from
definitions. The induction step is obtained from

$h_{\overline{M}_{p}^{i+1}(K,L)}=h_{\overline{M}_{p}(\overline{M}_{p}^{i}(K,L),\underline{M}_{-p}^{i}(K,L))}\leq
m_{p}(h_{\overline{M}_{p}^{i}(K,L)},h_{\underline{M}_{-p}^{i}(K,L)})\leq
m_{p}(\varphi_{i},\psi_{i})=\varphi_{i+1}$

and by Lemma 2.1 also

$\displaystyle h_{\underline{M}_{-p}^{i+i}(K,L)}$
$\displaystyle=h_{\underline{M}_{-p}(\overline{M}_{p}^{i}(K,L),\underline{M}_{-p}^{i}(K,L))}\leq
h_{\overline{M}_{-p}(\overline{M}_{p}^{i}(K,L),\underline{M}_{-p}^{i}(K,L))}$
$\displaystyle\leq
m_{-p}(h_{\overline{M}_{p}^{i}(K,L)},h_{\underline{M}_{-p}^{i}(K,L)})\leq
m_{-p}(\varphi_{i},\psi_{i})=\psi_{i+1}.$

Therefore, by (25)

$h_{G_{p}(K,L)}=\lim_{i\to\infty}h_{\overline{M}_{p}^{i}(K,L)}\leq\limsup_{i\to\infty}\varphi_{i}=m_{0}(h_{K},h_{L}),$

which implies $G_{p}(K,L)\subset\overline{M}_{0}(K,L)$ by the definition of
the latter set. (P7) further implies

$G_{p}(K,L)=(G_{p}(K^{\circ},L^{\circ}))^{\circ}\supset(\overline{M}_{0}(K^{\circ},L^{\circ}))^{\circ}=\underline{M}_{0}(K,L).$

For (P8), we observe that Remark 7.3 shows

$\underline{M}_{0}(K,L)\subset\underline{M}_{0}(\overline{M}_{0}(K,L),\underline{M}_{0}(K,L))\subset
G_{0}(K,L)\subset\overline{M}_{0}(\overline{M}_{0}(K,L),\underline{M}_{0}(K,L))\subset\overline{M}_{0}(K,L).$

Now, equality on the left-hand side in (P8) yields equality in

$\underline{M}_{0}(K,L)=\underline{M}_{-\infty}(\overline{M}_{0}(K,L),\underline{M}_{0}(K,L))\subset\underline{M}_{0}(\overline{M}_{0}(K,L),\underline{M}_{0}(K,L)),$

which by Theorem 3.2 shows $\underline{M}_{0}(K,L)=\overline{M}_{0}(K,L)$.
This in turn implies $n=1$ or that $K$ is a dilatate of $L$ by Theorem 3.1.
Equality on the right-hand side in (P8) can be handled analogously. Lastly, if
$n=1$ or $K$ is a dilatate of $L$, then Theorem 3.1 shows equality on both
sides in (P8). ∎

Lastly, (P9) is again a direct extension of the behavior of the geometric mean
for real numbers. It is what mainly sets $G_{0}$ apart from the other $G_{p}$,
$p\neq 0$, as for example [19, Appendix] shows that $G_{1}$ violates this
property.

###### Proof of (P9).

We have

$\overline{M}_{0}(\alpha K,\beta
L)=\bigcap_{a\in\mathbb{S}_{2}}H_{(a,\sqrt{\alpha\beta
h_{K}(a)h_{L}(a)})}^{\leq}=\bigcap_{a\in\mathbb{S}_{2}}\sqrt{\alpha\beta}\cdot
H_{(a,\sqrt{h_{K}(a)h_{L}(a)})}^{\leq}=\sqrt{\alpha\beta}\cdot\overline{M}_{0}(K,L)$

and by Lemma 2.1 also

$\underline{M}_{0}(\alpha K,\beta L)=(\overline{M}_{0}(\alpha^{-1}\cdot
K^{\circ},\beta^{-1}\cdot
L^{\circ}))^{\circ}=(\sqrt{\alpha^{-1}\beta^{-1}}\cdot\overline{M}_{0}(K^{\circ},L^{\circ}))^{\circ}=\sqrt{\alpha\beta}\cdot\underline{M}_{0}(K,L).$

Thus, it follows from induction that for all $i\in\mathbb{N}$

(28) $\displaystyle\overline{M}_{0}^{i}(\alpha K,\beta
L)=\sqrt{\alpha\beta}\cdot\overline{M}_{0}^{i}(K,L)$ $\displaystyle\text{ and
}\underline{M}_{0}^{i}(\alpha K,\beta
L)=\sqrt{\alpha\beta}\cdot\underline{M}_{0}^{i}(K,L)$

and therefore, by basic properties of Hausdorff limits, that (P9) holds true.
∎

Let us observe that this proof of (P9) cannot be reused for any $p\neq 0$. As
Magazinov’s example [19, Appendix] shows for $p=1$, this cannot be remedied
even with different proof ideas for general $p\in\mathbb{R}$.

Let us further point out that the above proof shows that the properties (P1),
(P4), and (P6) are automatically fulfilled for any function
$G\colon\mathcal{C}^{n}_{0}\times\mathcal{C}^{n}_{0}\to\mathcal{C}^{n}_{0}$
that satisfies the other six properties in Theorem 7.4. To our knowledge, it
is an open problem whether the remaining six properties already suffice to
uniquely characterize $G_{0}$ (cf. [18]) or whether additional properties are
needed to do so.

Next, we consider some additional properties that one would expect of a
geometric mean. They are concerned with various aspects of symmetrization and
measuring (a)symmetry. We again first state, then discuss, and finally proof
them.

######  Corollary 7.5.

Let $K,L\in\mathcal{C}^{n}_{0}$. Then for all $p\in\mathbb{R}\colon$

  1. (P10)

$G_{p}(K,-K)$ is $0$-symmetric.

  2. (P11)

If $K$ is Minkowski centered, then $d_{BM}(K,G_{p}(K,-K))=s(K)$.

  3. (P12)

If $K$ and $L$ are Minkowski centered, then
$s(G_{p}(K,L))\leq\max\\{s(K),s(L)\\}$, where equality implies that
$G_{p}(K,L)$ is Minkowski centered. In particular, if $K$ and $L$ are
$0$-symmetric, then so is $G_{p}(K,L)$.

It is natural to expect (P10) of any kind of (unweighted) mean of a convex
body and its negative if it should not be biased toward either of them.

(P11) is mostly expected due to the context given by Corollary 6.1. The latter
in particular shows for a Minkowski centered $K\in\mathcal{C}^{n}_{0}$ that
its $p$-symmetrizations according to the definition in (3), i.e. $p\neq 0$ and
$L=-K$, are equidistant from $K$ w.r.t. the Banach-Mazur distance. It seems
appropriate to demand that the 0-symmetrization should also have the same
distance from $K$ in this case.

Lastly, it is natural to expect a “mean” of 0-symmetric convex bodies to also
be 0-symmetric. (P12) generalizes this behavior to Minkowski centered bodies
in the sense that it would be somewhat counter-intuitive if a “mean” of
$K,L\in\mathcal{C}^{n}_{0}$ would be more asymmetric than both $K$ and $L$. It
should rather be at a structural midpoint between $K$ and $L$, and at least
satisfy the inequality in (P12).

Let us briefly discuss the assumption of Minkowski centeredness in (P12): All
$G_{p}$ depend on the translation of their arguments. The example below shows
that $G_{p}(K,L)$ does not need to be symmetric if $K$ and $L$ are symmetric
with their centers not at the origin, even if their centers coincide. We
therefore need to fix an appropriate translation of the bodies $K$ and $L$ to
obtain meaningful results even in this special case. In the context of
Minkowski asymmetry, requiring Minkowski centeredness is a rather intuitive
choice. It would, for example, restrict $K$ and $L$ to have their centers at
the origin if they are symmetric.

######  Example 7.6.

Let $n\geq 2$, $K\coloneqq 18\mathbb{B}_{1}+2u^{n}$ and
$L\coloneqq\operatorname{conv}(\\{5u^{n},-u^{n}\\}\cup\\{\pm
18u^{i}+2u^{n}:i\in[n-1]\\})$. Then $K$ and $L$ are $(2u^{n})$-symmetric.
However, $G_{p}(K,L)$ is not symmetric for any $p\in\mathbb{R}\colon$

By Lemma 2.2, the points $\sqrt{20\cdot 5}u^{n}=10u^{n},-\sqrt{16\cdot
1}u^{n}=-4u^{n},\pm 18u^{i}+2u^{n}$, $i\in[n-1]$, lie in the boundary of both
$\underline{M}_{0}(K,L)$ and $\overline{M}_{0}(K,L)$. By
$\underline{M}_{0}(K,L)\subset G_{p}(K,L)\subset\overline{M}_{0}(K,L)$, these
points are also boundary points of $G_{p}(K,L)$. Moreover,
$G_{p}(K,L)\subset\operatorname{conv}(K\cup L)=K$ implies that $\pm
18u^{1}+2u^{n}$ is the only point in $G_{p}(K,L)\cap H_{(\pm
u^{1},h_{G_{p}(K,L)}(\pm u^{1}))}$. Thus, the only possible symmetry center of
$G_{p}(K,L)$ is $2u^{n}$. However, $(2u^{n}-(-4u^{n}))=6u^{n}\neq
8u^{n}=(10u^{n}-2u^{n})$ implies that $G_{p}(K,L)$ is not
$(2u^{n})$-symmetric, either. Hence, $G_{p}(K,L)$ is asymmetric.

In the following, we essentially show that all three properties in Corollary
7.5 are automatically satisfied if the properties in Theorem 7.4 (except
possibly (P8) and (P9)) are fulfilled.

###### Proof of Corollary 7.5.

(P10) is a direct consequence of (P2) and (P5) by

$-G_{p}(K,-K)=G_{p}(-K,-(-K))=G_{p}(-K,K)=G_{p}(K,-K).$

(P11) follows immediately from (P8), (P10) and Corollary 6.1.

Lastly, (P3) and (P5) yield

$\displaystyle-G_{p}(K,L)$ $\displaystyle=G_{p}(-K,-L)\subset
G_{p}(s(K)K,s(L)L)$ $\displaystyle\subset G_{p}(\max\\{s(K),s(L)\\}\cdot
K,\max\\{s(K),s(L)\\}\cdot L)=\max\\{s(K),s(L)\\}\cdot G_{p}(K,L)$

and thus the inequality in (P12). It also shows the Minkowski centeredness in
case of equality. The claims about the case of 0-symmetric $K$ and $L$ are
immediate consequences of the above. ∎

The last property that we consider in this section is a strengthening of (P12)
for $G_{0}$, whose proof relies on (P9).

######  Corollary 7.7.

Let $K,L\in\mathcal{C}^{n}_{0}$ be Minkowski centered. Then

(29) $s(G_{0}(K,L))\leq\sqrt{s(K)s(L)},$

where equality implies that $G_{0}(K,L)$ is Minkowski centered. Moreover, for
all $s,t\in[1,n]$, there exist Minkowski centered bodies
$K^{\prime},L^{\prime}\in\mathcal{C}^{n}_{0}$ with $s(K^{\prime})=s$ and
$s(L^{\prime})=t$ such that $s(G_{0}(K^{\prime},L^{\prime}))=\sqrt{st}$.

Note that the following proof of tightness essentially shows that the upper
bound in (29) cannot be improved for any function
$G\colon\mathcal{C}^{n}_{0}\times\mathcal{C}^{n}_{0}\to\mathcal{C}^{n}_{0}$
that satisfies (P8).

###### Proof.

For the inequality, we use (P3), (P5), and (P9):

$-G_{0}(K,L)=G_{0}(-K,-L)\subset G_{0}(s(K)K,s(L)L)=\sqrt{s(K)s(L)}\cdot
G_{0}(K,L).$

We also see that $\sqrt{s(K)s(L)}=s(G_{0}(K,L))$ implies that $G_{0}(K,L)$ is
Minkowski centered.

To show tightness of the inequality, let $T^{n}\in\mathcal{C}^{n}_{0}$ be a
Minkowski centered $n$-simplex. For any $r\in[1,n]$, let $K_{r}\coloneqq
T^{n}\cap(-rT^{n})$. Then

$-\frac{r}{n}T^{n}\subset(rT^{n})\cap(-T^{n})=-K_{r}\subset(rT^{n})\cap(-r^{2}T^{n})=rK_{r}\subset
rT^{n}$

together with $s(T^{n})=n$ shows that $K_{r}$ is Minkowski centered with
$s(K_{r})=r$. Moreover, by Lemma 2.1, (28), (29) and (P8), there exists
$c\in\mathbb{R}^{n}$ with

$\displaystyle-\frac{\sqrt{st}}{n}T^{n}$
$\displaystyle=-\frac{\sqrt{st}}{n}\cdot\underline{M}_{0}(T^{n},T^{n})=-\underline{M}_{0}\left(\frac{s}{n}T^{n},\frac{t}{n}T^{n}\right)\subset-\underline{M}_{0}(K_{s},K_{t})\subset-
G_{0}(K_{s},K_{t})$ $\displaystyle\subset s(G_{0}(K_{s},K_{t}))\cdot
G_{0}(K_{s},K_{t})+c\subset
s(G_{0}(K_{s},K_{t}))\cdot\overline{M}_{0}(K_{s},K_{t})+c$
$\displaystyle\subset
s(G_{0}(K_{s},K_{t}))\cdot\overline{M}_{0}(T^{n},T^{n})+c=s(G_{0}(K_{s},K_{t}))\cdot
T^{n}+c.$

Thus, $s(T^{n})=n$ implies $s(G_{0}(K_{s},K_{t}))\geq\sqrt{st}$. ∎

######  Remark 7.8.

In closing the paper, let us again consider the approach of proving (22) via
finding a geometric mean
$G\colon\mathcal{C}^{n}_{0}\times\mathcal{C}^{n}_{0}\to\mathcal{C}^{n}_{0}$
that satisfies (23) and (24). Equality in (22) for $K,L\in\mathcal{C}^{n}_{0}$
would then require $G(K,L)=\overline{M}_{0}(K,L)$. If $G$ additionally
satisfies (P7), then also (P8) by (23), and equality in (22) for
$K,L\in\mathcal{C}^{n}_{0}$ would further imply
$G(K^{\circ},L^{\circ})=\underline{M}_{0}(K^{\circ},L^{\circ})$. Since
equality is known to hold in (22) if $K$ and $L$ are axis-parallel
$0$-symmetric boxes, we would obtain from Theorem 3.1 in the case of $n\geq 2$
and $K,L$ not being dilatates of each other that

(30) $\underline{M}_{0}(K,L)\subsetneq
G(K,L)=\overline{M}_{0}(K,L)\quad\text{and}\quad\underline{M}_{0}(K^{\circ},L^{\circ})=G(K^{\circ},L^{\circ})\subsetneq\overline{M}_{0}(K^{\circ},L^{\circ}).$

The equality case in (P8) thus shows that $G_{0}$ must violate (24), so
$G=G_{0}$ is not a valid choice. In general, it might not be impossible to
define a map $G$ as described above, but the behavior outlined in (30) seems
somewhat unnatural if we want to think of $G$ as a geometric mean as discussed
above.

It is worth mentioning in this context that the dual inequality to (22) is
proved in [23, Theorem 6.1]. The inequality there in particular yields

(31) $\forall
K,L\in\mathcal{C}^{n}_{0}\colon\operatorname{vol}(\underline{M}_{0}(K,L))\leq\sqrt{\operatorname{vol}(K)\cdot\operatorname{vol}(L)}.$

If a function
$G\colon\mathcal{C}^{n}_{0}\times\mathcal{C}^{n}_{0}\to\mathcal{C}^{n}_{0}$
that satisfies (23), (24), and (P7) exists, then the above argument would show
that equality in (22) for $K,L\in\mathcal{C}^{n}_{0}$ implies equality in (31)
for $K^{\circ}$ and $L^{\circ}$. To our knowledge, the characterization of the
equality case for either inequality is still open.

## References

  * [1] K. J. Böröczky, The Logarithmic Minkowski Conjecture and the $L_{p}$-Minkowski Problem. Harmonic Analysis and Convexity (2023), 83–118.
  * [2] K. J. Böröczky, E. Lutwak, D. Yang, G. Zhang, The log-Brunn-Minkowski Inequality. Adv. Math., Vol. 231 (2012), 1974–1997.
  * [3] R. Brandenberg, K. von Dichter, B. Gonzáles Merino, Tightening and Reversing the Arithmetic-Harmonic Mean Inequality for Symmetrizations of Convex Sets. Commun. Contemp. Math., Vol. 25, No. 9 (2023).
  * [4] R. Brandenberg, S. König, No Dimension-Independent Core-Sets for Containment Under Homothetics. Discrete Comput. Geom., Vol. 49, No. 1 (2013), 3–21.
  * [5] S. Chen, Y. Huang, Q. Li, J. Liu, The $L_{p}$-Brunn-Minkowski Inequality for $p<1$. Adv. Math., Vol. 368 (2020), 107166.
  * [6] G. Crasta, I. Fragalà, On a Geometric Combination of Functions Related to Prékopa-Leindler Inequality. Mathematika, Vol. 69, No. 2 (2023), 482–507.
  * [7] V. P. Fedotov, Geometric Mean of Convex Sets. J. Sov. Math., Vol. 10, No. 3 (1978), 488–491.
  * [8] W. J. Firey, $p$-Means of Convex Bodies. Math. Scand., Vol. 10 (1962), 17–24.
  * [9] W. J. Firey, Polar Means of Convex Bodies and a Dual to the Brunn-Minkowski Theorem. Canad. J. Math, Vol. 13 (1961), 444–453.
  * [10] B. Grünbaum, Measures of Symmetry for Convex Sets. Proc. Sympos. Pure Math., Vol. 7 (1963), 233–270.
  * [11] F. Grundbacher, On Fundamental Functionals of Convex Geometry: New, Refined, and Corrected Results. Master’s Thesis, Technical University of Munich (2023).
  * [12] T. Kobos, M. Varivoda, On the Banach-Mazur Distance in Small Dimensions. arXiv preprint arXiv:2305.06427.
  * [13] A. V. Kolesnikov, E. Milman, Local $L^{p}$-Brunn-Minkowski Inequalities for $p<1$. Mem. Amer. Math. Soc., Vol. 277 (2022), 1360.
  * [14] J. D. Lawson, Y. Lim, The Geometric Mean, Matrices, Metrics, and More. Amer. Math. Monthly, Vol. 108, No. 9 (2001), 797–812.
  * [15] E. Lutwak, The Brunn-Minkowski-Firey Theory I: Mixed Volumes and the Minkowski Problem. J. Differential Geom., Vol 38, No. 1 (1993), 131–150.
  * [16] E. Lutwak, The Brunn-Minkowski-Firey Theory II: Affine and Geominimal Surface Areas. Adv. Math., Vol. 118, No. 2 (1996), 244–294.
  * [17] E. Lutwak, D. Yang, G. Zhang, The Brunn-Minkowski-Firey Inequality for Nonconvex Sets. Adv. in Appl. Math., Vol. 48, No. 2 (2012), 407–413.
  * [18] V. Milman, L. Rotem, “Irrational” Constructions in Convex Geometry. Algebra Anal., Vol. 29, No. 1 (2017), 222–236.
  * [19] V. Milman, L. Rotem, Non-standard Constructions in Convex Geometry: Geometric Means of Convex Bodies. Convexity and Concentration, The IMA Volumes in Mathematics and its Applications, Vol. 161, Springer New York (2017), 361–390.
  * [20] V. Milman, L. Rotem, Weighted Geometric Means of Convex Bodies. Contemp. Math., Vol. 733 (2019), 233–249.
  * [21] E. Putterman, Equivalence of the Local and Global Versions of the $L^{p}$-Brunn-Minkowski Inequality. J. Funct. Anal., Vol. 280, No. 9 (2021), 108956.
  * [22] L. Rotem, Banach Limit in Convexity and Geometric Means for Convex Bodies. Electron. Res. Announc. Math. Sci., Vol. 23 (2016), 41–51.
  * [23] C. Saroglou, More on Logarithmic Sums of Convex Bodies. Mathematika, Vol. 62, No. 3 (2016), 818–841.
  * [24] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Second expanded ed., Cambridge University Press (2014).
  * [25] A. E. Taylor, General Theory of Functions and Integration. Blaisdell Publishing Co. (1965).
  * [26] D. Xi, G. Leng, Dar’s Conjecture and the log-Brunn-Minkowski Inequality. J. Differential Geom., Vol. 103, No. 1 (2016), 145–189.

René Brandenberg – Technical University of Munich, Department of Mathematics,
Germany.
<EMAIL_ADDRESS>
Florian Grundbacher – Technical University of Munich, Department of
Mathematics, Germany.
<EMAIL_ADDRESS>