## Section: New Results

### Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry

#### Isometries of the Hilbert geometry

Participant : Cormac Walsh.

In a collaboration with Bas Lemmens (Kent University, UK), we have been studying the Hilbert geometry in finite dimensions, especially its horofunction boundary and isometry group. The book chapter [117] contains a survey of this work. However, the infinite dimensional case is also interesting, and has been used as a tool for many years in non-linear analysis. Despite this, very little is known about the geometry of these spaces when the dimension is infinite.

An example of a problem in which we are interested is the following. In finite dimension it is known that a Hilbert geometry is isometric to a normed space if and only if it is a simplex. We have shown [118] that, more generally, a Hilbert geometry is isometric to a Banach space if and only if it is the cross-section of a positive cone, that is, the cone of positive continuous functions on some compact topological space. To solve this problem we found it useful to study the horofunction boundary in the infinite-dimensional case.

We are continuing to study similar problems in relation to this topic in collaboration with Bas Lemmens of the University of Kent.

#### Volume growth in the Hilbert geometry

Participant : Cormac Walsh.

In a collaboration with Constantin Vernicos of Université Montpellier 2, we are investigating how the volume of a ball in a Hilbert geometry grows as its radius increases. Specifically, we are studying the volume entropy

where $B(x,r)$ is the ball with center $x$ and radius $r$, and $Vol$ denotes some notion of volume, for example, the Holmes–Thompson or Busemann definitions. Note that the entropy does not depend on the particular choice of $x$, nor on the choice of the volume. It is known that the hyperbolic space, or indeed any Hilbert geometry with a ${C}^{2}$-smooth boundary of strictly positive curvature, has entropy $n-1$, where $n$ is the dimension, and it has recently been proved that this is the maximal entropy possible for Hilbert geometries of the given dimension.

Constantin Vernicos has shown that, in dimension 2 and 3, the volume entropy
of a Hilbert geometry on a convex body is equal to exactly twice the
*approximability* of the body, that is, the power of $1/\u03f5$
governing the growth of the number of vertices needed to approximate the body
by a polytope within $\u03f5$, as $\u03f5$ decreases.

Studying polytopal Hilbert geometries,
we have demonstrated [53] a close relation
between the volume and the number of *flags* of the polytope,
more precisely, that the volume of large balls is asymptotically proportional
to the number of flags.
This suggested to us defining a new notion of approximability using
flags rather than vertices. We have shown [53]
that the volume entropy
of a Hilbert geometry on a convex body is equal to exactly twice
this *flag-approximability* in all dimensions.
This implies in particular
that the volume entropy of a convex body is equal to that of its dual.

#### The set of minimal upper bounds of two matrices in the Loewner order

Participant : Nikolas Stott.

A classical theorem of Kadison shows that the space of symmetric matrices equipped with the Loewner order is an anti-lattice, meaning that two matrices have a least upper bound if and only if they are comparable. In [52], we refined this theorem by characterizing the set of minimal upper bounds: we showed that it is homeomorphic to the quotient space $O\left(p\right)\setminus O(p,q)/O\left(q\right)$, where $O(p,q)$ denotes the orthogonal group associated to the quadratic form with signature $(p,q)$, and $O\left(p\right)$ denotes the standard $p$th orthogonal group.

#### Checking the strict positivity of Kraus maps is NP-hard

Participant : Stéphane Gaubert.

In collaboration with Zheng Qu (now with HKU, Hong Kong), I studied several decision problems arising from the spectral theory of Kraus maps (trace preserving completely positive maps), acting on the cone of positive semidefinite matrices. The latter appear in quantum information. We showed that checking the irreducibility (absence of non-trivial invariant face of the cone) and primitivity properties (requiring the iterates of the map to send the cone to its interior) can be checked in polynomial time, whereas checking positivity (whether the map sends the cone to its interior) is NP-hard. In [17], we studied complexity issues related to Kraus maps, and showed in particular that checking whether a Kraus map sends the cone to its interior is NP-hard.