arXiv research

The study shows how strictly convex domains in Euclidean spaces are rigid.

Universal inequalities found for Laplacian eigenvalues on convex domains.

Flat subsets in Euclidean buildings are contained within apartments.

Universal inequalities for Laplacian eigenvalues on convex domains.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.

Solves Christoffel-Minkowski problem for capillary convex bodies in Euclidean half-space.

Study optimizes perimeter in convex domains with anisotropic constraints.

Paper proves rigidity of convex hypersurfaces in various spaces.

This paper extends a 3D result to higher dimensions for manifolds with positive curvature.

New methods optimize functions on hyperbolic and spherical spaces, matching Euclidean rates up to logarithmic factors.

The paper explores centroids and static equilibrium points in non-Euclidean geometries.

Study on convex capillary hypersurfaces with Lp curvature in half-space.

Paper revisits DP-SCO in Euclidean and $\ell_p^d$ spaces, focusing on constrained and bounded sets.

New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.

The paper studies convexity of products of squared Euclidean distances.

New algorithms optimize convex functions with high-order derivatives.

New upper bound for Neumann Laplacian eigenvalues on convex domains.

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…

For a Euclidean building $X$ of type $A_{2}$, we classify the 0-dimensional subbuildings $A$ of $\partial_{T}X$ that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a triple (of points of $A$) is (essentially) sufficient. To prove this, we construct n…

The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condit…

New inequalities for convex hypersurfaces in various spaces.

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting met…

The study proves non-orientable surfaces can map to a torus.

Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

Riemannian algorithms converge at Euclidean rates for geodesically convex-concave problems.

The paper proves a theorem about mean curvature in Euclidean and hyperbolic spaces.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

New interpretation of discrete conformality using polyhedral convex hulls.

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

The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.

Convex hypersurfaces in curved spaces bound convex regions.

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometri…

The paper proves new inequalities in hyperbolic space using Euclidean methods.

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds o…

The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.

This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is that, under good conditions on Fock-Goncharov generalized shear parameters, non-Arch…

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.

The paper proves the exact number of singular points in the intersection of convex shapes.

Develops iso-Riemannian optimization for data manifolds.

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …

Adaptive step-size improves optimization in complex geometries.

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.

We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.

We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.