In a letter to Felix Klein published in Mathematische Annalen 1895 (see here), Hilbert generalized the Cayley-Klein model of hyperbolic geometry by defining a metric on the interior of a convex body in the following way: given distinct interior points $x$ and $y$ let $a$ and $b$ be the two points obtained by intersecting the boundary of the convex body and the straight line defined by $x$ and $y$. Assuming, by renaming $a$ and $b$ if necessary, that the euclidean distance from $x$ to $b$ is greater than the distance from $y$ to $b$, we set

$$ d(x,y) = \frac{1}{2} \ln\left(\frac{|x-b|}{|y-b|}\frac{|y-a|}{|x-a|}\right) . $$

This metric has two remarkable properties: (1) straight lines are geodesics and (2) projective transformation that send the body into itself are distance non-increasing.

For a nice elementary account of these geometries (in French ...) see this note by Ludovic Marquis.

Question 1. What is the infimum of the volumes (volume = Hausdorff $n$-dimensional measure) of all unit balls in all $n$-dimensional Hilbert geometries?

Question 2. What is the supremum of the volumes of all unit balls in all $n$-dimensional Hilbert geometry?

Remark. The infimum is necessarily smaller than or equal to the volume of the Euclidean unit ball of dimension $n$. To see this, notice that when the convex body is a simplex, the Hilbert geometry is isometric to a certain normed space (de la Harpe) and by a theorem of Busemann the Hausdorff volume of the unit ball of any normed space is equal to the volume of the Euclidean unit ball. I guess that the loosely stated fact that Hilbert geometries are non-positively curved makes it likely that the infimum is actually the volume of the Euclidean unit ball of dimension $n$. On the other had, the loosely stated fact that hyperbolic geometry is the most "hyperbolic" of the Hilbert geometries would seem to imply that the supremum is the volume of the hyperbolic unit ball of dimension $n$.

Edited April 17, 2014. A short argument I learned from this paper of Ludovic Marquis shows that the infimum and the supremum in the questions are attained. In particular, the volume of every unit ball in every Hilbert geometry is greater than some fixed number $c > 0$ and is less than some fixed number $C < \infty$.

Consider the space $C(n)$ of all proper convex bodies in projective $n$-space and let $$ C_p(n) := \{(K,x) \in C(n) \times \mathbb{RP}^n : x \hbox{ is in the interior of } K \} $$ be the space of pointed convex bodies. The topology on $C(n)$ is taken to be the topology given by the Hausdorff metric for closed subsets of $\mathbb{RP}^n$ and this topology is used to give a topology to the space of pointed convex bodies in the obvious way.

A theorem of Benzecri (but I think it can be obtained almost immediately from an older result of Macbeath) states that the projective group acts properly and co-compactly on $C_p(n)$. Since the map that takes a pair $(K,x)$ to the volume of the unit ball centered at $x$ in the Hilbert geometry determined by $K$ is continuous and invariant under the action, it follows that this volume attains its extremal values.

These questions arose in a conversation with Constantin Vernicos (and in retrospect I see that he actually hinted this argument to me a while back).


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