arXiv research

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …

A new projection method for convex optimization reduces computation costs.

We study a notion of convex cocompactness for discrete subgroups of the projective general linear group acting (not necessarily irreducibly) on real projective space, and give various characterizations. A convex cocompact group in this sense need not be word hyperbolic, but we show that it still has some of the good pr…

Entropy rigidity proven for 3D and higher convex projective manifolds.

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

Study of hyperbolic directions in convex projective geometry.

Geodesic flow mixing on convex projective manifolds proven.

A convex projective surface is the quotient of a properly convex open $Ω$ of $\mathbb{P}(\R)$ by a discret subgroup $Γ$ of $\mathrm{SL}_3(\R)$. We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if $Ω$ is not a triangle then …

An open convex set in real projective space is called divisible if there exists a discrete group of projective automorphisms which acts co-compactly. There are many examples of such sets and a theorem of Benoist implies that many of these examples are strictly convex, have $C^1$ boundary, and have word hyperbolic divid…

We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth l…

Unique entropy measure found for convex projective manifolds.

Entropy study of geodesic flow on convex projective surfaces.

Paper proposes a new clustering model that preserves cluster recovery with fewer dimensions.

Cyclic projections in Hadamard spaces can be irregular, unlike in Hilbert spaces.

Space curves with convex projections evolve smoothly until shrinking to a point.

The paper characterizes groups acting on real projective spaces.

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…

In this paper, we consider the problem of learning high-dimensional tensor regression problems with low-rank structure. One of the core challenges associated with learning high-dimensional models is computation since the underlying optimization problems are often non-convex. While convex relaxations could lead to polyn…

Characterizes convex cocompact actions in projective space with dynamical properties.

Characterizes holonomies of convex projective cusps.

We prove that elliptic tubes over properly convex domains of the real projective space are C-convex and complete Kobayashi-hyperbolic. We also study a natural construction of complexification of convex real projective manifolds.

For any sequence of properly convex domains in the real projective plane such that the zeros of Pick differentials have bounded multiplicity and get further and further apart, we determined all Hausdorff limit domains that one can obtain after normalizing each member of the sequence by a projective transformation. We t…

Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.

Proves EGF representations in specific geometric contexts.

We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture…

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…

Study shows non-symmetric convex sets have full boundary limits.

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times …

Symplectic coordinates found on projective structures on orbifolds.

In this paper we study the degeneration of convex real projective structures on bordered surfaces.

Study on projective orbifolds with ends and their deformation theory.

Proof that convex structures on manifolds are open and closed.

Study real projective structures on a specific Coxeter orbifold.

A dissertation on scalable projection-free optimization methods.

The paper explores volume product and slicing conjectures using convex body deformations.

Properly convex manifolds with generalized cusps have irreducible holonomy.

Paper links set derivatives to its orthogonal projections.

For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of $M_m$ are Gromov-hyperbolic …

The paper proves convexity results for a specific type of Lie groups.

For convex real projective manifolds we prove an analogue of the higher rank rigidity theorem of Ballmann and Burns-Spatzier.

New algorithm reduces adaptive regret without projections.

Characterizes Coxeter groups with convex cocompact representations in projective space.

The paper studies the correlation of Hilbert lengths for convex projective surfaces.

The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.

In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective $n$-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…

3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.

In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We find a necessary and sufficient condition that ensures a positive answer to both questions and suggest an algorithm of construction of the body. Also we fin…

In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…