# Weak Finsler Strutures and the Funk Metric

Athanase Papadopoulos  A. Papadopoulos, Institut de Recherche Mathématique
Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084
Strasbourg Cedex - France<EMAIL_ADDRESS>and  Marc Troyanov
M. Troyanov, Section de Mathématiques, École Polytechnique Fédérale de
Lausanne, 1015 Lausanne - Switzerland<EMAIL_ADDRESS>
(Date: April 3, 2008)

###### Abstract.

We discuss general notions of metrics and of Finsler structures which we call
_weak metrics_ and _weak Finsler structures_. Any convex domain carries a
canonical weak Finsler structure, which we call its _tautological weak Finsler
structure_. We compute distances in the tautological weak Finsler structure of
a domain and we show that these are given by the so-called _Funk weak metric_.
We conclude the paper with a discussion of geodesics, of metric balls and of
convexity properties of the Funk weak metric.

AMS Mathematics Subject Classification: 52A, 53C60, 58B20 Keywords: Finsler
structure, weak metric, Funk weak metric.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on convex geometry
  3. 3 The notion of weak metric
  4. 4 Weak length spaces
  5. 5 Weak Finsler structures
  6. 6 The tautological weak Finsler structure
  7. 7 The Funk weak metric
  8. 8 On the geometry of the Funk weak metric

##  1\. Introduction

A _weak metric_ on a set is a function defined on pairs of points in that set
which is nonnegative, which can take the value $\infty$, which vanishes when
the two points coincide and which satisfies the triangle inequality. Compared
to an ordinary metric, a weak metric can thus degenerate and take infinite
values. Besides, it need not be symmetric. This is a very general notion which
turns out to be useful in various situations. The terminology “weak metric” is
due to Ribeiro [20], but the notion can be at least traced back to the work of
Hausdorff (see [14]). In the paper [18], a number of natural weak metrics are
discussed. In the present paper, we are mostly interested in a class of weak
metrics that is related to convex geometry and to a general notion of Finsler
structures on manifolds.

A basic construction in convex geometry is the notion of _Minkowski norm_ ,
which associates to any convex set containing the origin in a vector space $V$
a translation-invariant homogenous weak metric on $V$. Finsler geometry is an
extension of this construction to an arbitrary manifold. We define a _weak
Finsler strucure_ on a differentiable manifold to be a field of convex sets on
that manifold. More precisely, a weak Finsler strucure is a subset of the
tangent space of the manifold whose intersection with each fiber is an convex
set containing the origin. The Minkowski norm in each tangent space of a
manifold endowed with a weak Finsler structure gives rise to a function
defined on the total space of the tangent bundle. We call this function the
_Lagrangian_ of the weak Finsler structure. Integrating this Lagrangian on
piecewise smooth curves in the manifold defines a length structure and thus a
notion of distance on the manifold. This distance is generally a weak metric.

A case of special interest is when the manifold is a convex domain $\Omega$ in
$\mathbb{R}^{n}$ and when the weak Finsler structure is obtained by
replicating at each point of $\Omega$ the domain $\Omega$ itself. We call this
the _tautological_ weak Finsler structure, and we study some of its basic
properties in the present paper. More precisely, we first give a formula for
the distance between two points. It turns out that this distance coincides
with the metric introduced by P. Funk in [13]. We then study the geometry of
balls and the geodesics in the Funk weak metric.

Modern references on Finsler geometry include [9], [2], [3], and [1]. One of
Herbert Busemann’s major ideas, expressed in [5], [6], [7] and [8] is that
Finsler geometry should be developed without local coordinates and without the
use of differential calculus. This paper brings some results in that
direction.

##  2\. Preliminaries on convex geometry

In this section, we recall a few notions in convex geometry that will be used
in the sequel.

Given a convex subset $\Omega$ of $\mathbb{R}^{n}$, we shall denote
$\overline{\Omega}$ its closure, $\stackrel{{\scriptstyle o}}{{\Omega}}$ its
interior, and
$\partial\Omega=\overline{\Omega}\setminus\stackrel{{\scriptstyle
o}}{{\Omega}}$ its boundary.

Let $\Omega\subset\mathbb{R}^{n}$ be a (not necessarily open) convex set and
let $x$ be a point in $\Omega$.

######  Definition 2.1.

The _radial function of $\Omega$ with respect to $x$_ is the function
$r_{\Omega,x}:\mathbb{R}^{n}\to\mathbb{R}_{+}\cup\\{\infty\\}$ defined by

$r_{\Omega,x}(\xi)=\sup\\{t\in\mathbb{R}\ |\ (x+t\xi)\in\Omega\\}.$

######  Definition 2.2.

The _Minkowski function of $\Omega$ with respect to $x$_ is the function
$p_{\Omega,x}:\mathbb{R}^{n}\to\mathbb{R}_{+}\cup\\{\infty\\}$ defined by

$p_{\Omega,x}(\xi)=\frac{1}{r_{\Omega,x}(\xi)}.$

Classically, the Minkowski function is associated to an open convex subset
$\Omega$ of $\mathbb{R}^{n}$ containing the origin $0$, and taking $x=0$. This
function is sometimes called the _Minkowski weak norm_ of the convex (see e.g.
[10], [17], [21] and [22]).

We also recall that for any convex set $\Omega$ in $\mathbb{R}^{n}$, there
exists a well-defined smallest affine subspace $L$ of $\mathbb{R}^{n}$
containing $\Omega$, and that the intersection of $\Omega$ with $L$ has
nonempty interior in $L$. We denote by $\mathrm{RelInt}(\Omega)$ this
interior, called the _relative interior of the convex set $\Omega$_.

The following proposition collects a few basic properties of the Minkowski
function. In particular, Property (8) tells us that we can reconstruct the
relative interior of $\Omega$ from the Minkowski function of $\Omega$ at any
point. The proofs are easy.

######  Proposition 2.3.

Let $\Omega$ be a convex subset of $\mathbb{R}^{n}$. For every $x$ in $\Omega$
and for every $\xi$ and $\eta$ in $\mathbb{R}^{n}$, we have

  1. (1)

$p_{\Omega,x}(\xi)=\inf\\{t\geq 0\ |\ \xi\in t(\Omega-x)\\}$. (Here,
$\Omega-x$ denotes the Minkowski sum of $\Omega$ and $-x$.)

  2. (2)

If the ray $\\{x+t\xi\ |\ t\geq 0\\}$ is contained in $\Omega$, then
$p_{\Omega,x}(\xi)=0$.

  3. (3)

$p_{\Omega,x}(\lambda\xi)=\lambda p_{\Omega,x}(\xi)$ for $\lambda\geq 0$.

  4. (4)

$p_{\Omega,x}(\xi+\eta)\leq p_{\Omega,x}(\xi)+p_{\Omega,x}(\eta)$.

  5. (5)

The Minkowski function $p_{\Omega,x}$ is convex.

  6. (6)

If $x$ is in $\stackrel{{\scriptstyle o}}{{\Omega}}$, then $p_{{\Omega},x}$ is
continuous.

  7. (7)

If ${\Omega}$ is closed, then ${\Omega}=\\{y\in\mathbb{R}^{n}\ |\
y=x+\xi,p_{{\Omega},x}(\xi)\leq 1\\}$.

  8. (8)

$\mathrm{RelInt}(\Omega)=\\{y=x+\xi|\ p_{{\Omega},x}(\xi)<1\\}$.

  9. (9)

If $\Omega_{1}=\mathrm{RelInt}(\Omega)$, then
$p_{{\Omega_{1}},x}=p_{{\Omega},x}$.

In some cases, we can give explicit formulas for the Minkowski function
$p_{{\Omega},x}$. For instance, the Minkowski function of the closed ball
$B=B(0,R)$ in $\mathbb{R}^{n}$ of radius $R$ and center $0$ with respect to
any point $x$ in $B$ is given by

$p_{B,x}(\xi)=\frac{\sqrt{\langle\xi,x\rangle^{2}+(R^{2}-|x|^{2})|\xi|^{2}}+\langle\xi,x\rangle}{(R^{2}-|x|^{2})}.$

The Minkowski function of a half-space
$H=\\{x\in\mathbb{R}^{n}\,\big{|}\,\langle\nu,x\rangle\leq s\\}$, where $\nu$
is a vector in $\mathbb{R}^{n}$ (which is orthogonal to the hyperplane
bounding $H$) and where $s$ is a real number, with respect to a point $x$ in
$H$, is given by

$p_{H,x}(\xi)=\max\left(\frac{\langle\nu,\xi\rangle}{s-\langle\nu,x\rangle},0\right).$

We shall use this formula later on in this paper. We also recall the
following:

######  Definition 2.4 (Support hyperplane).

Let $\Omega$ be a nonempty subset of $\mathbb{R}^{n}$. An affine hyperplane
$A$ in $\mathbb{R}^{n}$ is called a _support hyperplane_ for ${\Omega}$ if the
relative interior of ${\Omega}$ is contained in one of the two closed half-
spaces bounded by $A$ and if $\overline{\Omega}\cap A\not=\emptyset$.

If $A$ is a support hyperplane for ${\Omega}$ and if $x$ is a point in
$\overline{{\Omega}}\cap A$, then $A$ is called a _support hyperplane for
${\Omega}$ at $x$._ When $\Omega\subset\mathbb{R}^{2}$, then $A$ is called a
support line.

Suppose now that $\Omega$ a convex subset of $\mathbb{R}^{n}$. It is known
that any point on the boundary of ${\Omega}$ is contained in at least one of
its support hyperplanes (see e.g. [10] p. 20). The intersection of ${\Omega}$
with any of its support hyperplanes is a convex set which is nonempty if
${\Omega}$ is closed. This intersection is not always reduced to a point.

We recall the notion of a strictly convex subset in $\mathbb{R}^{n}$, and
before that we note the following classical proposition:

######  Proposition 2.5.

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$. Then, the following
are equivalent:

  1. (1)

$\partial\Omega$ does not contain any nonempty open affine segment;

  2. (2)

each support hyperplane of $\Omega$ intersects $\partial\Omega$ in exactly one
point;

  3. (3)

support hyperplanes at distinct points of $\partial\Omega$ are distinct;

  4. (4)

any linear function on $\mathbb{R}^{n}$ has exactly one maximum on
$\partial\Omega$.

######  Definition 2.6 (Strictly convex subset).

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$. Then, $\Omega$ is
said to be _strictly convex_ if one (or, equivalently, all) the properties of
Proposition 2.5 are satisfied.

##  3\. The notion of weak metric

######  Definition 3.1.

A _weak metric_ on a set $X$ is a function $\delta:X\times
X\to\mathbb{R}_{+}\cup\\{\infty\\}$ satisfying

  1. (1)

$\delta(x,x)=0$ for all $x$ in $X$;

  2. (2)

$\delta(x,z)\leq\delta(x,y)+\delta(y,z)$ for all $x$, $y$ and $z$ in $X$.

We say that such a weak metric $\delta$ is _symmetric_ if
$\delta(x,y)=\delta(y,x)$ for all $x$ and $y$ in $X$; that it is _finite_ if
$\delta(x,y)<\infty$ for every $x$ and $y$ in $X$; that $\delta$ is _strongly
separating_ if we have the equivalence

$\min(\delta(x,y),\delta(y,x))=0\iff x=y;$

and that $\delta$ is _weakly separating_ if we have the equivalence

$\max(\delta(x,y),\delta(y,x))=0\iff x=y.$

We recall that the notion of weak metric already appears in the work of
Hausdorff (cf. [14], in which Hausdorff defines asymmetric distances on
various sets of subsets of a metric space).

######  Definition 3.2 (Geodesic).

Let $(X,\delta)$ be a weak metric space and let $I\subset\mathbb{R}$ be an
interval. We say that a map $\gamma:I\to X$ is _geodesic_ if for every
$t_{1}$, $t_{2}$ and $t_{3}$ in $I$ satisfying $t_{1}\leq t_{2}\leq t_{3}$ we
have

$\delta(\gamma(t_{1}),\gamma(t_{2}))+\delta(\gamma(t_{2}),\gamma(t_{3}))=\delta(\gamma(t_{1}),\gamma(t_{3})).$

Weak metrics were extensively studied by Busemann, cf. [5], [6], [7] & [8]. A
basic example of a weak metric defined on a convex set in $\mathbb{R}^{n}$ is
the following:

######  Example 3.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a convex set such that
$0\in\overline{\Omega}$ and let $p(\xi)=p_{\Omega,0}(\xi)=\inf\\{t>0\ |\
\xi\in t\,\Omega\\}$ be the Minkowski weak norm centered at $0$ of $\Omega$.
Then, the function
$\delta:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}_{+}\cup\\{\infty\\}$
defined by

$\delta(x,y)=p(y-x)$

is a weak metric on $\mathbb{R}^{n}$. For this weak metric, we have the
following equivalences:

  1. (1)

$\delta$ is finite $\iff$ $0\in\stackrel{{\scriptstyle o}}{{\Omega}}$;

  2. (2)

$\delta$ is symmetric $\iff$ $\Omega=-\Omega$;

  3. (3)

$\delta$ is strongly separating $\iff$ $\Omega$ is bounded;

  4. (4)

$\delta$ is weakly separating $\iff$ $\Omega$ does not contain any Euclidean
line.

The weak metric on $\mathbb{R}^{n}$ defined in Example 3.3 is called the
_Minkowski weak metric associated to $\Omega$_. The associated weak metric
space $(\mathbb{R}^{n},\delta)$ is called a _weak Minkowski space_.

##  4\. Weak length spaces

Let $X$ be a set and let $\Gamma$ be a groupoid of paths in $X$. Concatenation
of paths is denoted by the symbol $*$. The inverse $\gamma^{-1}$ of a path
$\gamma:[a,b]\to X$ is the path obtained by pre-composing $\gamma$ with the
unique affine sense-reversing self-homeomorphism of $[a,b]$.

######  Definition 4.1.

A _weak length structure_ on $(X,\Gamma)$ is a function
$\ell:\Gamma\to\mathbb{R}_{+}\cup\\{\infty\\}$ which satisfies the following
properties:

  1. (1)

_Invariance under reparametrization_ : if $[a,b]$ and $[c,d]$ are intervals of
$\mathbb{R}$, if $\gamma:[a,b]\to X$ is a path in $X$ that belongs to $\Gamma$
and if $f:[c,d]\to[a,b]$ is an increasing homeomorphism such that $\gamma\circ
f\in\Gamma$, then $\ell(\gamma)=\ell(\gamma\circ f)$.

  2. (2)

_Additivity_ : for every $\gamma_{1}$ and $\gamma_{2}$ in $\Gamma$, we have
$\ell(\gamma_{1}*\gamma_{2})=\ell(\gamma_{1})+\ell(\gamma_{2})$.

A weak length structure $\Gamma$ is said to be _reversible_ if for every
$\gamma$ in $\Gamma$, $\gamma^{-1}$ is also in $\Gamma$ and we have
$\ell(\gamma^{-1})=\ell(\gamma)$.

A weak length structure $\Gamma$ is said to be _separating_ if we have the
equivalence: $\ell(\gamma)=0\iff$ $\gamma$ is a unit in $\Gamma$ (i.e.
$\gamma$ is a constant path).

Let $(X,\Gamma,\ell)$ be a set equipped with a groupoid of paths and with a
weak length structure. We set

(4.1)
$\displaystyle\delta_{\ell}(x,y)=\inf_{\gamma\in\Gamma_{x,y}}\ell(\gamma),$

where

$\Gamma_{x,y}=\\{\gamma\in\Gamma\ |\ \gamma\textrm{ joins $x$ to $y$ }\\}.$

It is easy to see that the function $\delta_{\ell}$ is a weak metric on $X$.

######  Definition 4.2.

Let $(X,\Gamma,\ell)$ be a set equipped with a groupoid of paths and with a
weak length structure. The weak metric $\delta_{\ell}$ defined in (4.1) is
called the _weak metric associated to the weak length structure $\ell$_. A
_weak length metric space_ is a weak metric space $X$ obtained from such a
triple $(X,\Gamma,\ell)$ by equipping $X$ with the associated weak metric
$\delta_{\ell}$.

##  5\. Weak Finsler structures

We introduce a general notion of Finsler structure, which we call _weak
Finsler structure_ , and which can be considered as an infinitesimal notion of
weak length structure.

######  Definition 5.1.

Let $M$ be a $C^{1}$ manifold and let $TM$ be its tangent bundle. A _weak
Finsler structure_ on $M$ is a subset $\widetilde{\Omega}\subset TM$ such that
for each $x$ in $M$, the subset $\Omega_{x}=\widetilde{\Omega}\cap T_{x}M$ of
the tangent space $T_{x}M$ of $M$ at $x$ is convex and contains the origin.

We provide the set of all weak Finsler structures on $M$ with the order
relation $\preceq$ defined as follows:

$\widetilde{\Omega_{1}}\preceq\widetilde{\Omega_{2}}\ \Leftrightarrow\
\widetilde{\Omega_{1}}\supset\widetilde{\Omega_{2}}.$

######  Examples 5.2.

In the following examples, $M$ is a $C^{1}$ manifold.

  1. (1)

$\widetilde{\Omega}=TM$ is a weak Finsler structure, which we call the
_minimal_ weak Finsler structure.

  2. (2)

$\widetilde{\Omega}=M\subset TM$, embedded as the zero section, is a weak
Finsler structure which we call the _maximal_ weak Finsler structure.

  3. (3)

If $\widetilde{\Omega}$ and $\widetilde{\Omega^{\prime}}$ are two Finlser
structures on $M$, then
$\widetilde{\Omega}\cap\widetilde{\Omega^{\prime}}\subset TM$ is also a
Finsler structure.

  4. (4)

If $\widetilde{\Omega}$ and $\widetilde{\Omega^{\prime}}$ are two Finlser
structures on $M$, then, taking the union of the Minkowski sums
${\Omega_{x}}+\Omega^{\prime}_{x}$ of the convex sets in each tangent space
$T_{x}M$, we obtain the _Minkowski sum Finsler structure_
$\widetilde{\Omega}+\widetilde{\Omega^{\prime}}\subset TM$.

  5. (5)

If $\omega$ is a differential 1-form on $M$, then

$\widetilde{\Omega}_{\omega}=\\{(x,\xi)\in TM\ |\ \omega_{x}(\xi)\leq 1\\}$

and

$\widetilde{\Omega}_{|\omega|}=\\{(x,\xi)\in TM\ |\ |\omega_{x}|(\xi)\leq
1\\}$

are weak Finsler structures on $M$.

  6. (6)

If $\omega$ and $\omega^{\prime}$ are two 1-froms on $M$, then
$\max(\omega,\omega^{\prime})$ defines a weak Finlser structure on $M$.

  7. (7)

If $\widetilde{\Omega}$ is a weak Finlser structure on $M$ and if $N\subset M$
is a $C^{1}$ submanifold, then $\widetilde{\Omega}_{N}=\widetilde{\Omega}\cap
TN$ is a weak Finlser structure on $N$, called the weak Finsler structure
_induced_ by the embedding $N\subset M$.

  8. (8)

If $\widetilde{\Omega}$ is a weak Finlser structures on $M$, if $N$ is a
$C^{1}$ manifold and if $f:N\to M$ is a $C^{1}$ map, then
$(Tf)^{-1}(\widetilde{\Omega})\subset TN$ is a Finsler structure on $N$. We
denote it by $f^{*}(\widetilde{\Omega})$ and call it the _pull back_ of
$\widetilde{\Omega}$ by the map $f$.

######  Definition 5.3 (Lagrangian).

The _Lagrangian_ of a weak Finlser structure $\widetilde{\Omega}$ on a $C^{1}$
manifold $M$ is the function on the tangent bundle $TM$ whose restriction to
each tangent space $T_{x}$ is the Minkowski function of $\Omega_{x}$. It is
thus defined by

$p(x,\xi)=p_{\widetilde{\Omega}}(x,\xi)=\inf\\{t\ |\
t^{-1}\xi\in\Omega_{x}\\}.$

The quantity $p(x,\xi)$ is also called the _Finsler norm_ of the vector
$(x,\xi)$ relative to the given weak Finlser structure.

######  Example 5.4.

Let $g$ be a Riemannian metric on $M$, let $\omega$ is a differential 1-form
and let $\mu$ be a smooth function on $M$ satisfying $|\mu\omega_{x}|<1$ at
every point $x$ in $M$. Then, $p=\sqrt{g}+\mu\omega$ is the Lagrangian of a
Finsler structure on $M$. Such a Finlser structure is usually called a
_Randers metric_ on $M$, and it has applications on physics (cf. e.g. [3]
§11.3, and see also [4] for the relation of this metric with the Zermelo
navigation problem.)

######  Lemma 5.5.

Let $\widetilde{\Omega}$ be a weak Finlser structure on $M$. Assume that $M$
(considered as a subset of $TM$ \- the zero section) is contained in the
interior of $\widetilde{\Omega}\subset TM$. Then the associated Lagrangian
$p:TM\to\mathbb{R}$ is upper semi-continuous.

###### Proof.

The hypothesis implies that for every $x$ in $M$, the interior of each convex
set $\Omega_{x}=\widetilde{\Omega}\cap T_{x}M\subset T_{x}M$ is nonempty.
Therefore, the usual interior and the relative interior of $\Omega_{x}$
coincide. Property (9) of Proposition 2.3 implies then that the Lagrangian of
$\widetilde{\Omega}$ coincides with the Lagrangian of its interior
$\mathrm{Int}\left(\widetilde{\Omega}\right)$.

One may therefore assume without loss of generality that
$\widetilde{\Omega}\subset TM$ is an open set, and in particular

$\widetilde{\Omega}=\\{(x,\xi)\in TM\,\big{|}\,\ p(x,\xi)<1\\}$

(see Proposition 2.3 (8)). Now for any $t\in\mathbb{R}$, the sublevel set
$\\{p(x,\xi)<t\\}$ is either empty (when $t\leq 0$) or it is homothetic to the
open set $\widetilde{\Omega}\subset TM$ (when $t>0$). In any case, it is an
open subset of $TM$, and $p:TM\to\mathbb{R}$ is therefore upper semi-
continuous.

∎

######  Proposition 5.6.

Let $\widetilde{\Omega}$ be a Finsler structure on a $C^{1}$ manifold $M$ and
let $p_{\widetilde{\Omega}}:TM\to\mathbb{R}$ be the associated Lagrangian.
Then,

  1. (1)

for every $x$ in $M$, the function $\xi\mapsto p(x,\cdot)$ is a weak norm on
$T_{x}M$;

  2. (2)

if $\widetilde{\Omega^{\prime}}\subset TM$ is another Finsler structure on
$M$, with associated Lagrangian $p_{\widetilde{\Omega^{\prime}}}$, then we
have the equivalence

$\widetilde{\Omega}\preceq\widetilde{\Omega^{\prime}}\iff
p_{\widetilde{\Omega}}\leq p_{\widetilde{\Omega^{\prime}}},$

  3. (3)

$p_{\widetilde{\Omega}}:TM\to\mathbb{R}$ is Borel-measurable.

###### Proof.

The first two assertions are easy to check and we only prove the last one. If
$M$ is contained in the interior of $\widetilde{\Omega}\subset TM$, then, by
Lemma 5.5, the Lagrangian $p$ is upper semi-continuous and therefore Borel
measurable. In the general case, $M$ is contained in $\widetilde{\Omega}$ but
not necessarily in its interior. We consider a decreasing sequence

$TM\preceq\widetilde{\Omega}_{1}\preceq\widetilde{\Omega}_{2}\preceq\cdots\preceq\widetilde{\Omega}$

of weak Finsler structures such that $M$ is contained in the interior of
$\widetilde{\Omega}_{j}\subset TM$ for every $j\in\mathbb{N}$ and

$\widetilde{\Omega}=\bigcap_{j=1}^{\infty}\widetilde{\Omega}_{j}$

We then have $p_{\widetilde{\Omega}_{1}}\leq
p_{\widetilde{\Omega}_{2}}\leq\cdots\leq p_{\widetilde{\Omega}}$ and

$p_{\widetilde{\Omega}}=\sup_{j}p_{\widetilde{\Omega}_{j}}=\lim_{j\to\infty}p_{\widetilde{\Omega}_{j}}$

Therefore $p_{\widetilde{\Omega}}$ is the limit of a sequence of Borel
measurable functions and is thus Borel measurable.

∎

We shall say that the Finlser structure $\widetilde{\Omega}$ is _smooth_ if
$p$ is smooth.

######  Definition 5.7 (The weak length structure associated to a weak Finsler
structure).

Let $M$ be a $C^{1}$ manifold equipped with a weak Finlser structure
$\widetilde{\Omega}$ with Lagrangian $p$. There is an associated weak length
structure on $M$, defined by taking $\Gamma$ to be the groupoid of piecewise
$C^{1}$ paths, and defining, for each $\gamma:[a,b]\to M$ in $\Gamma$,

(5.1) $\ell(\gamma)=\int_{a}^{b}p(\gamma(t),\dot{\gamma}(t))dt.$

######  Remark 5.8.

In Equation (5.1), $\gamma$ and $\dot{\gamma}$ are continuous, and since $p$
is Borel-measurable, the map $t\mapsto p(\gamma(t),\dot{\gamma}(t))$ is
nonnegative and measurable. Therefore, the Lebesgue integral is well defined.

##  6\. The tautological weak Finsler structure

In this section, $\Omega$ is an open convex subset of $\mathbb{R}^{n}$. We
shall use the natural identification
$T\Omega\simeq\Omega\times\mathbb{R}^{n}$.

######  Definition 6.1 (The tautological weak Finsler structure).

The _tautological weak Finsler structure_ on $\Omega$ is the weak Finsler
structure $\widetilde{\Omega}\subset T\Omega$ defined by

$\widetilde{\Omega}=\\{(x,\xi)\in\Omega\times\mathbb{R}^{n}\ |\
x\in\Omega\text{ and }x+\xi\in\Omega\\}.$

This structure is called “tautological” because the fibre over each point $x$
of $\Omega$ is the set $\Omega$ itself (with the origin at $x$).

The proof of next proposition follows easily from the definitions.

######  Proposition 6.2.

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$ equipped with its
tautological weak Finsler structure $\widetilde{\Omega}$. Then, for every $x$
in $\Omega$, the Finsler norm of any tangent vector $\xi$ at $x$ is given by
$p_{\Omega,x}(\xi)$, where $p_{\Omega,x}$ is the Minkowski function of
$\Omega$ with respect to $x$.

Given an open convex subset $\Omega$ of $\mathbb{R}^{n}$, we denote by
$d_{\Omega}$ the weak length metric associated to the tautological weak
Finsler structure on $\Omega$. This weak metric is thus defined by

(6.1)
$d_{\Omega}(x,y)=\inf_{\gamma\in\Gamma_{x,y}}\int_{\gamma}p(\gamma(t),\dot{\gamma}(t))dt.$

where $\Gamma_{x,y}$ is the set of piecewise $C^{1}$ paths joining $x$ to $y$.

######  Lemma 6.3.

Let $\Omega$ and $\Omega^{\prime}$ be two convex open subsets of
$\mathbb{R}^{n}$ satisfying $\Omega\subset\Omega^{\prime}$, then
$d_{\Omega^{\prime}}\leq d_{\Omega}$.

In the rest of this paper, we shall use the following notations: For $x$ and
$y$ in $\mathbb{R}^{n}$, we denote by $|x-y|$ their Euclidean distance. Given
two distinct points $x$ and $y$ in $\Omega$, $R(x,y)$ denote the Euclidean ray
starting at $x$ and passing through $y$. In the case where
$R(x,y)\not\subset\Omega$ we set $a^{+}=a^{+}(x,y)=R(x,y)\cap\partial\Omega$.

######  Theorem 6.1.

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$ equipped with its
tautological weak Finsler structure. Then, for every $x$ and $y$ in $\Omega$,
the Euclidean segment connecting $x$ and $y$ is of minimal length, and the
associated weak metric on $\Omega$ is given by

$\displaystyle
d_{\Omega}(x,y)=\begin{cases}\displaystyle\log\frac{|x-a^{+}|}{|y-a^{+}|}&\text{
if }x\not=y\text{ and }R(x,y)\not\subset\Omega\\\ 0&\text{
otherwise}.\end{cases}$

###### Proof.

As before, we let $d_{\Omega}$ denote the weak metric defined by the
tautological weak Finsler structure on $\Omega$. We also denote by
$\ell(\gamma)$ the length of a path $\gamma$ for the tautological weak Finsler
weak length structure.

The proof of the theorem is done in four steps.

_Step 1.—_ Suppose that $R(x,y)\subset\Omega$. Consider the linear path
$\gamma:[0,|x-y|]\to\Omega$ defined by

(6.2) $\gamma(t)=x+t\frac{y-x}{|y-x|}.$

The derivative of the path $\gamma$ is the constant vector

$\dot{\gamma}(t)=\frac{y-x}{|y-x|}.$

Therefore, $\displaystyle
p_{\widetilde{\Omega}}(\gamma(t),\dot{\gamma}(t))=\frac{1}{|y-x|}p_{\widetilde{\Omega}}(\gamma(t),y-x)$,
which is equal to 0 since $R(x,y)\subset\Omega$.

Now the path $\gamma$ has length zero and satisfies $\gamma(0)=x$ and
$\gamma(|y-x|)=y$. Therefore $d_{\Omega}(x,y)=0$.

In the rest of this proof, we suppose that $R(x,y)\not\subset\Omega$.

_Step 2.—_ We show that for every distinct points $x$ and $y$ in $\Omega$ and
for every Euclidean segment $\gamma$ joining $x$ to $y$, we have

(6.3) $\displaystyle
d_{\Omega}(x,y)\leq\ell(\gamma)=\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

Using the radial function $r_{\Omega,x}$ introduced in §2, we can write

$a^{+}=a(x,y-x)=x+r_{\Omega,x}(y-x)\cdot(y-x).$

To compute the Finsler length of the Euclidean segment $[x,y]$, we parametrize
it as the path $\gamma$ defined in (6.2).

For $0\leq t\leq|x-y|$, let $r(t)=|x-\gamma(t)|$. Then,
$r(t)=r_{\Omega,x}(\gamma(t),\dot{\gamma}(t))$, and it is easy to see that

$r(t)=|x-a^{+}|-t.$

Then, we have $r^{\prime}(t)=-1$ and therefore

$\ell(\gamma)=\int_{0}^{|y-x|}\frac{dt}{r(t)}=-\int_{0}^{|y-x|}\frac{r^{\prime}(t)dt}{r(t)}=-\log\big{(}r(t)\big{)}\Big{|}_{t=0}^{t=|y-x|}=\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

This gives the desired inequality (6.3).

_Step 3.—_ We complete the proof of the theorem in the particular case where
$\Omega$ is a half-space. By the invariance of the tautological Finsler
structure under the group of affine transformations, it suffices to consider
the case where $\Omega$ is the half-space $H\subset\mathbb{R}^{n}$ defined by
the equation

$H=\\{x\in\mathbb{R}^{n}\,\big{|}\,\langle\nu,x\rangle\leq s\\},$

for some vector $\nu$ in $\mathbb{R}^{n}$ (which is orthogonal to the
hyperplane bounding $H$) and for some $s$ in $\mathbb{R}$. Recall that the
Minkowski function associated to $H$ is given by the formula

$p_{H}(x,\xi)=\max\left\\{\frac{\langle\nu,\xi\rangle}{s-\langle\nu,x\rangle},0\right\\}.$

Consider now an arbitrary piecewise $C^{1}$ path $\alpha:[0,1]\to H$ such that
$x=\alpha(0)$ and $y=\alpha(1)$. Then,

$\ell(\alpha)=\int_{0}^{1}\max\left\\{\frac{\langle\nu,\dot{\alpha}(t)\rangle}{s-\langle\nu,\alpha(t)\rangle},0\right\\}dt\geq\int_{0}^{1}\frac{\langle\nu,\dot{\alpha}(t)\rangle}{s-\langle\nu,\alpha(t)\rangle}dt.$

We have

$\frac{\langle\nu,\dot{\alpha}(t)\rangle}{s-\langle\nu,\alpha(t)\rangle}=-\frac{d}{dt}\big{(}\log\big{(}s-\langle\nu,\alpha(t)\rangle\big{)}\big{)}.$

Therefore,

$\ell(\alpha)\geq-\log\big{(}s-\langle\nu,\alpha(1)\rangle\big{)}+\log\big{(}s-\langle\nu,\alpha(0)\rangle\big{)}=\log\frac{s-\langle\nu,x\rangle}{s-\langle\nu,y\rangle}.$

Now we note that

$s-\langle\nu,x\rangle=s-\langle\nu,x-a^{+}\rangle-\langle\nu,a^{+}\rangle=\langle
x-a^{+},-\nu\rangle=\langle\nu,a^{+}-x\rangle.$

Likewise,

$s-\langle\nu,y\rangle=\langle\nu,a-y\rangle.$

Thus, we obtain

$\ell(\alpha)\geq\log\frac{\langle\nu,a^{+}-x\rangle}{\langle\nu,a^{+}-y\rangle}.$

Now using the fact that the three points $x,y,a^{+}$ are aligned in that order
and that $\nu$ is not parallel to the vector $x-y$, we easily see that

$\frac{\langle\nu,a-x\rangle}{\langle\nu,a-y\rangle}=\frac{|x-a^{+}|}{|y-a^{+}|},$

which gives

$\ell(\alpha)\geq\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

Since $\alpha$ was arbitrary, we have

$d_{H}(x,y)\geq\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

Combining this inequality and the inequality (6.3), we obtain, in the case
where $\Omega=H$ is a half-space,

$d_{H}(x,y)=\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

In particular any Euclidean segment is length minimizing.

_Step 4.—_ Now we prove the proposition for a general open convex set
$\Omega$.

Let $x$ and $y$ be two elements in $\Omega$ and consider the Euclidean ray
$R(x,y)$.

By hypothesis, we have $L\not\subset\Omega$, and as before, we set
$a^{+}=R(x,y)\cap\partial\Omega$. We let $A$ denote a support hyperplane to
$\Omega$ through $a^{+}$, and we let $H$ be the open half-space containing
$\Omega$ and whose boundary is equal to $A$. Using Lemma 6.3 and Step 3, we
have

$d_{\Omega}(x,y)\geq d_{H}(x,y)=\log\frac{|x-a^{+}|}{|y-a^{+}|}.$

Combining this with the inequality (6.3) we obtain
$d_{\Omega}(x,y)=\log\frac{|x-a^{+}|}{|y-a^{+}|}$. The argument also proves
that any Euclidean segment $\gamma$ is length minimizing. This completes the
proof of Theorem 6.1.

∎

##  7\. The Funk weak metric

In this and the following section, we give a quick overview of the Funk weak
metric, of its geodesics, of its balls and of its topology.

The Funk weak metric is a nice example of a weak metric, and a geometric study
of this weak metric is something which seems missing in the literature. We
study this weak metric in more detail in [19].

In this section, $\Omega$ is a nonempty open convex subset of
$\mathbb{R}^{n}$. We use the notations $a^{+}$, $R(x,y)$, etc. established in
the preceding section.

######  Definition 7.1 (The Funk weak metric).

The _Funk weak metric_ of $\Omega$, denoted by $F_{\Omega}$, is defined, for
$x$ and $y$ in $\Omega$, by the formula

$\displaystyle
F_{\Omega}(x,y)=\begin{cases}\displaystyle\log\frac{|x-a^{+}|}{|y-a^{+}|}&\text{
if }x\not=y\text{ and }R(x,y)\not\subset\Omega\\\ 0&\text{
otherwise}.\end{cases}$

Observe that Theorem 6.1 says that the Funk weak metric is the weak metric
associated to the tautological Finsler structure in $\Omega$. In particular
the triangle inequality is verified. Another proof of the triangle inequality
is given in [23] p. 85. This proof is not trivial and uses arguments similar
to those of the classical proof of the triangle inequality for the Hilbert
metric, as given by D. Hilbert in [16].

If $\Omega=\mathbb{R}^{n}$, then $F\equiv 0$. We shall henceforth assume that
$\Omega\not=\mathbb{R}^{n}$ whenever we shall deal with the Funk weak metric
of an nonempty open convex subset $\Omega$ of $\mathbb{R}^{n}$.

The Funk weak metric is always unbounded. Indeed, if $x$ is any point in
$\Omega$ and if $x_{n}$ is any sequence of points in that space converging to
a point on $\partial\Omega$ (convergence here is with respect to the Euclidean
metric), then $F_{\Omega}(x,x_{n})\to\infty$. Notice that on the other hand
$F_{\Omega}(x_{n},x)$ is bounded.

######  Example 7.2 (The upper half-plane).

Let $\Omega=H\subset\mathbb{R}^{2}$ be the upper half-plane, that is,

$H=\\{(x_{1},x_{2})\in\mathbb{R}^{2}\ |\ x_{2}>0\\}.$

Then, for $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ in $H$, we have

$F_{H}(x,y)=\max\left\\{\log\frac{x_{2}}{y_{2}},0\right\\}.$

The following three propositions are easy consequences of the definitions and
they will be used below. We take $\Omega$ to be again a nonempty open subset
of $\mathbb{R}^{n}$.

######  Proposition 7.3.

Let $\Omega^{\prime}\subset\Omega$ be the intersection of $\Omega$ with an
affine subspace of $\mathbb{R}^{n}$, and suppose that
$\Omega^{\prime}\not=\emptyset$. Then, $F_{\Omega^{\prime}}$ is the weak
metric induced by $F_{\Omega}$ on $\Omega^{\prime}$.

######  Proposition 7.4.

In the case where $\Omega$ is bounded, the Funk weak metric $F_{\Omega}$ is
strongly separating, and we have the following equivalences:

(7.1) $F_{\Omega}(x,x_{n})\to 0\iff F_{\Omega}(x_{n},x)\to 0\iff|x-x_{n}|\to
0.$

######  Proposition 7.5.

Let $\Omega_{1}$ and $\Omega_{2}$ be two open convex subsets of
$\mathbb{R}^{n}$. Then,

$F_{\Omega_{1}\cap\Omega_{2}}=\max\left\\{F_{\Omega_{1}},F_{\Omega_{1}}\right\\}.$

##  8\. On the geometry of the Funk weak metric

In this section, we study the geodesics, and then, the geometric balls of the
Funk weak metric.

######  Proposition 8.1.

Let $x$, $y$ and $z$ be three points in $\Omega$ lying in that order on a
Euclidean line. Then, we have $F(x,y)+F(y,z)=F(x,z)$.

This results follows from Theorem 6.1, but it is also quite simple to prove it
directly.

###### Proof.

We can assume that the three points are distinct, otherwise the proof is
trivial. We have $R(x,y)\subset\Omega\iff R(x,z)\subset\Omega\iff
R(y,z)\subset\Omega$, and this holds if and only if the three quantities
$F(x,y)$, $F(y,z)$ and $F(x,z)$ are equal to 0. Thus, the conclusion also
holds trivially in this case. Therefore, we can assume that
$R(x,y)\not\subset\Omega$. In this case, we have
$a^{+}(x,y)=a^{+}(x,z)=a^{+}(y,z)$. Denoting this common point by $a^{+}$, we
have

$\frac{|x-a^{+}|}{|y-a^{+}|}\frac{|y-a^{+}|}{|z-a^{+}|}=\frac{|x-a^{+}|}{|z-a^{+}|},$

which implies

$\log\frac{|x-a^{+}|}{|y-a^{+}|}+\log\frac{|y-a^{+}|}{|z-a^{+}|}=\log\frac{|x-a^{+}|}{|z-a^{+}|},$

which completes the proof. ∎

######  Corollary 8.2.

The Euclidean segments in $\Omega$ are geodesic segments for the Funk weak
metric on $\Omega$.

Since the open set $\Omega$ is convex, Corollary 8.2 implies that
$(\Omega,F_{\Omega})$ is a geodesic weak metric space (any two points can be
joined by a geodesic segment). It also says that $(\Omega,F_{\Omega})$ is a
Desarguesian space in the sense of H. Busemann (see [7]).

Notice that in general, the Euclidean segments are not the only geodesic
segments for a Funk weak metric. In fact, the following proposition implies
that there exist other types of geodesic segments in $\Omega$, provided there
exists a Euclidean segment of nonempty interior contained in the boundary of
$\Omega$.

######  Proposition 8.3.

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$ such that
$\partial\Omega$ contains a Euclidean segment $[p,q]$ and let $x$ and $z$ be
two points in $\Omega$ such that $R(x,z)\cap[p,q]\not=\emptyset$. Let
$\Omega^{\prime}$ be the intersection of $\Omega$ with the affine subspace of
$\mathbb{R}^{n}$ spanned by $\\{x\\}\cup[p,q]$. Then, for any point $y$ in
$\Omega^{\prime}$ satisfying $R(x,y)\cap[p,q]\not=\emptyset$ and
$R(y,z)\cap[p,q]\not=\emptyset$, we have $F(x,y)+F(y,z)=F(x,z)$.

Figure 1.

###### Proof.

It suffices to work in the space $\Omega^{\prime}$. Let $x^{\prime}$,
$y^{\prime}$ and $z^{\prime}$ denote the feet of the perpendiculars from $x$
and $z$ respectively on the Euclidean line joining the points $p$ and $q$ (see
Figure 1). Let $b=R(x,z)\cap[p,q]$. Since the triangles $bxx^{\prime}$ and
$bzz^{\prime}$ are similar, we have

$F(x,z)=\log\frac{|x-b|}{|z-b|}=\log\frac{|x-x^{\prime}|}{|z-z^{\prime}|}.$

Similar formulas hold for $F(x,y)$ and $F(y,z)$. Therefore,

$\displaystyle F(x,z)$
$\displaystyle=\log\frac{|x-x^{\prime}|}{|z-z^{\prime}|}$
$\displaystyle=\log\left(\frac{|x-x^{\prime}|}{|y-y^{\prime}|}\frac{|y-y^{\prime}|}{|z-z^{\prime}|}\right)$
$\displaystyle=\log\left(\frac{|x-x^{\prime}|}{|y-y^{\prime}|}\right)+\log\left(\frac{|y-y^{\prime}|}{|z-z^{\prime}|}\right)$
$\displaystyle=F(x,y)+F(y,z).$

∎

######  Remark 8.4.

By taking limits of polygonal paths, we can easily construct, from Proposition
8.3, smooth paths which are not Euclidean paths and which are geodesic for the
Funk weak metric.

######  Proposition 8.5.

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n}$. Let $x$ and $z$ be
two distinct points in $\Omega$ such that
$R(x,z)\cap\partial\Omega\not=\emptyset$ and such that at the point
$b=R(x,z)\cap\partial\Omega$, there is a support hyperplane whose intersection
with $\partial\Omega$ is reduced to $b$. Let $y$ be a point in $\Omega$ such
that the three points $x,y,z$ in $\Omega$ do not lie on the same affine line.
Then, $F(x,z)<F(x,y)+F(y,z)$.

###### Proof.

To prove the proposition, we work in the affine plane spanned by $x$, $y$ and
$z$ and therefore we can assume without loss of generality that $n=2$.

We assume that the intersection points of $R(x,y)$ and $R(y,z)$ with
$\partial\Omega$ are not empty, and we let $a$ and $c$ be respectively these
points. From the hypothesis, there is a support line of $\Omega$ (which we
call $D$) at $b$ whose intersection with $\partial\Omega$ is reduced to the
point $b$.

For the proof, we distinguish three cases.

Figure 2.

Case 1.— The two rays $R(x,y)$ and $R(y,z)$ intersect the line $D$ (see Figure
2).

Let $a^{\prime}$ and $c^{\prime}$ be respectively these intersection points.
Note that the three points $a^{\prime}$, $b$ and $c^{\prime}$ are in that
order on $D$. By reasoning with projections on the line $D$ and arguing as we
did in the proof of Proposition 8.3, we have

$\frac{|x-b|}{|z-b|}=\frac{|x-a^{\prime}|}{|y-a^{\prime}|}\frac{|y-c^{\prime}|}{|z-c^{\prime}|}.$

Since we have

$\frac{|x-a^{\prime}|}{|y-a^{\prime}|}<\frac{|x-a|}{|y-a|}$

and

$\frac{|y-c^{\prime}|}{|z-c^{\prime}|}<\frac{|y-c|}{|z-c|},$

we obtain

$\frac{|x-b|}{|z-b|}<\frac{|x-a|}{|y-a|}\frac{|y-c|}{|z-c|}$

which gives, by taking logarithms, $F(x,z)<F(x,y)+F(y,z)$.

Case 2.— The ray $R(x,y)$ intersects $D$ and the ray $R(y,z)$ does not
intersect $D$ (Figure 3). We let as before $a^{\prime}$ denote the point
$R(x,y)\cap D$.

Let $D^{\prime}$ be the Euclidean line passing through $z$ and parallel to
$D$. The hypotheses in the case considered imply that the line $D^{\prime}$
intersects the segment $[x,y]$. Let $y^{\prime}$ be this intersection point.
The point $y^{\prime}$ is contained in $\Omega$.

We have, as in Case 1,

$\displaystyle F(x,z)=\log\frac{|x-b|}{|z-b|}$

and

$F(x,y)=\log\frac{|x-a|}{|y-a|}>\log\frac{|x-a^{\prime}|}{|y-a^{\prime}|}.$

Now we have

$\frac{|x-b|}{|z-b|}=\frac{|x-a^{\prime}|}{|y^{\prime}-a^{\prime}|}<\frac{|x-a^{\prime}|}{|y-a^{\prime}|},$

that is, $F(x,z)<F(x,y)$, which implies the desired result.

Figure 3.

Case 3.— The ray $R(x,y)$ does not intersect the line $D$. This case can be
treated as Case 2, and we have in this case $F(x,y)<F(y,z)$, which implies the
desired result.

∎

The following is a direct consequence of Proposition 8.5.

######  Corollary 8.6.

Let $\Omega$ be an open bounded strictly convex subset of $\mathbb{R}^{n}$ and
let $x$, $y$ and $z$ be three points in $\Omega$ that are not contained in an
affine segment. Then, $F(x,z)<F(x,y)+F(y,z)$.

######  Corollary 8.7.

Let $\Omega$ be an open bounded strictly convex subset of $\mathbb{R}^{n}$.
Then, the affine segments in $\Omega$ are the only geodesic segments for the
Funk weak metric of $\Omega$.

###### Proof.

This follows from the previous Corollary and Corollary 8.2, which says that
the affine segments are geodesic segments for the Funk weak metric.

∎

We recall that a subset $Y$ in a (weak) metric space $X$ is said to be
_geodesically convex_ if for any two points $x$ and $y$ in $Y$, any geodesic
segment in $X$ joining $x$ and $y$ is contained in $Y$.

######  Corollary 8.8.

Let $\Omega$ be an open bounded strictly convex subset of $\mathbb{R}^{n}$ and
let $\Omega^{\prime}$ be a subset of $\Omega$. Then, $\Omega^{\prime}$ is
convex with respect to the affine structure of $\mathbb{R}^{n}$ if and only if
$\Omega^{\prime}$ is a geodesically convex subset of $\Omega$ with respect to
the Funk metric $F_{\Omega}$.

######  Remark 8.9.

Note the formal analogy between Corollary 8.7 and the following well known
result on the geodesic segments of a Minkowski metric on $\mathbb{R}^{n}$: if
the unit ball of a Minkowski metric is strictly convex, then the only geodesic
segments of this metric are the affine segments.

We now consider spheres and balls in a Funk weak metric space $(\Omega,F)$. As
this weak metric is non-symmetric, we have to distinguish between right and
left spheres, and we use the following notations. For any point $x$ in
$\Omega$ and any nonnegative real number $\delta$, we set

$\circ$ $B(x,\delta)=\\{y\in\Omega\ |\ F_{\Omega}(x,y)<\delta\\}$ (the _right
open ball of center $x$ and radius $\delta$_);

$\circ$ $B^{\prime}(x,\delta)=\\{y\in\Omega\ |\ F_{\Omega}(y,x)<\delta\\}$
(the _left open ball of center $x$ and radius $\delta$_);

$\circ$ $S(x,\delta)=\\{y\in\Omega\ |\ F_{\Omega}(x,y)=\delta\\}$ (the _right
sphere of center $x$ and radius $\delta$_);

$\circ$ $S^{\prime}(x,\delta)=\\{y\in\Omega\ |\ F_{\Omega}(y,x)=\delta\\}$
(the _left sphere of center $x$ and radius $\delta$_).

In [6] p. 20, H. Busemann discusses topologies for general weak metric spaces.
In the case of a genuine metric space, the open balls are used to define the
topology of that space. In general, the collections of left and of right open
balls in a weak metric space generate two different topologies. For the Funk
weak metric, we have the following

If $\Omega$ is a bounded convex open set of $\mathbb{R}^{n}$ equipped with its
Funk weak metric; then, the collections of left and of right open balls are
sub-bases of the same topology on $\Omega$, and this topology coincides with
the topology induced from the inclusion of $\Omega$ in $\mathbb{R}^{n}$.

In the case where the convex open set $\Omega$ is unbounded, the left and the
right open balls of the Funk weak metric are always noncompact. In the next
proposition, we study these balls in the case where $\Omega$ is bounded. We
recall that a convex subset of $\mathbb{R}^{n}$ is unbounded if and only if it
contains a Euclidean ray.

######  Proposition 8.10.

Let $\Omega$ be a bounded convex open subset of $\mathbb{R}^{n}$, let $x$ be a
point in $\Omega$ and let $\delta$ be a nonnegative real number. Then,

  1. (1)

The right sphere $S(x,\delta)$ is convex as a subset of $\mathbb{R}^{n}$, and
it is compact. Furthermore, this sphere is the image of $\partial\Omega$ by
the Euclidean homothety $\sigma$ of center $x$ and factor $(1-e^{-\delta})$.

  2. (2)

The left sphere $S^{\prime}(x,\delta)$ is convex as a subset of
$\mathbb{R}^{n}$, and it is equal to the intersection with $\Omega$ of the
image of $\partial\Omega$ by the Euclidean homothety of center $x$ and of
factor $(e^{\delta}-1)$, followed by the Euclidean central symmetry of center
$x$. The sphere $S^{\prime}(x,\delta)$ is not necessarily compact.

###### Proof.

Let $y$ be a point in $\Omega$ and let us set, as before,
$a^{+}=R(x,y)\cap\partial\Omega$. We have the following equivalences:

$y\in
S(x,\delta)\iff\log\frac{|x-a^{+}|}{|y-a^{+}|}=\delta\iff\displaystyle\frac{|x-a^{+}|}{|y-a^{+}|}=e^{\delta},$

which is easily seen to be equivalent to $|y-x|=|x-a^{+}|(1-e^{-\delta})$.
From this fact Property (1) follows easily.

To prove Property (2), let $a^{-}=R(y,x)\cap\partial\Omega$. We have the
following equivalences:

$\log\frac{|y-a^{-}|}{|x-a^{-}|}=\delta\iff|y-a^{-}|=e^{\delta}|x-a^{-}|,$

which is also equivalent to

$|y-x|=(e^{\delta}-1)|x-a^{-}|.$

Thus, $y\in S^{\prime}(x,\delta)$ if and only if $y$ is in the intersection of
$\Omega$ with the image $\sigma(\partial\Omega)$ of $\partial\Omega$ by the
Euclidean homothety with center $x$ and of factor $(e^{\delta}-1)$, followed
by the Euclidean central symmetry of center $x$. This intersection is convex
as a subset of $\mathbb{R}^{n}$ but it is not necessarily a compact subset of
$(\Omega,F)$. Thus, $S^{\prime}(x,\delta)$ is compact if and only if
$\sigma(\partial\Omega)$ is contained in $\Omega$. ∎

We note the following “local-implies-global” property of Funk weak metrics.
The meaning of the statement is clear, and it follows directly from
Proposition 8.10 (1).

######  Corollary 8.11.

We can reconstruct the boundary $\partial\Omega$ of $\Omega$ from the local
geometry at any point of $\Omega$.

######  Corollary 8.12.

Let $\Omega$ be a bounded open strictly convex subset of $\mathbb{R}^{n}$.
Then, the left and right open balls of $\Omega$ are geodesically convex with
respect to the Funk weak metric $F_{\Omega}$.

###### Proof.

This follows from Proposition 8.10 and from Corollary 8.8. ∎

We also deduce from Proposition 8.10 that for any $x$ and $x^{\prime}$ in
$\Omega$ and for any two positive real numbers $\delta$ and $\delta^{\prime}$,
the right spheres $S(x,\delta)$ and $S(x^{\prime},\delta^{\prime})$ are
homothetic.

Thus, for instance, if $\Omega$ is the interior of a Euclidean sphere
(respectively, of a Euclidean ellipsoid) in $\mathbb{R}^{n}$, then any right
sphere $S(x,\delta)$ is a Euclidean sphere (respectively, an ellipsoid).

Note that the proof of Proposition 8.10 shows that for a fixed $x$, any two
right spheres $S(x,\delta)$ and $S(x,\delta^{\prime})$ are homothetic by a
Euclidean homothety of center $x$, but that in general, a homothety which
sends a sphere $S(x,\delta)$ to a sphere $S(x^{\prime},\delta^{\prime})$ does
not necessarily send the center $x$ of $S(x,\delta)$ to the center
$x^{\prime}$ of $S(x^{\prime},\delta^{\prime})$. One can see this fact on the
following example: Let $\Omega$ be an open Euclidean disk in $\mathbb{R}^{n}$,
and let us take $x$ to be the Euclidean center of that disk. Then, by
symmetry, for any $\delta>0$, the right sphere $S(x,\delta)$ is a Euclidean
sphere whose Euclidean and whose metric centers are both at $x$. Now let
$x^{\prime}$ be a point which is close to the boundary of $\Omega$. Obviously,
the Euclidean homothety that sends $\partial\Omega$ to $S(x^{\prime},\delta)$
does not send the center of $\partial\Omega$ to the (Funk-)geometric center of
the sphere $S(x^{\prime},\delta)$ (recall that the center of this homothety is
the point $x$). Now taking a composition of two homotheties, we obtain a
Euclidean homothety that sends the geometric sphere $S(x,\delta)$ to the
geometric sphere $S(x^{\prime},\delta)$, and that does not preserve the
geometric centers of these spheres.

######  Remark 8.13.

The property for a weak metric on a subset $\Omega$ of $\mathbb{R}^{n}$ that
all the right spheres are homothetic is also shared by the metrics induced by
Minkowski weak metrics on $\mathbb{R}^{n}$.

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