arXiv research

The paper studies extremal hypersurfaces in ellipsoids using centro-affine geometry.

The paper proves an inequality and describes a curve flow in centro-affine geometry.

The paper studies a specific centro-affine invariant hypersurface flow in R^(n+1).

The paper matches features in images using centro-affine invariants and heat flow.

Employing a centro-affine flow on smooth convex bodies, we generate new centro-affine differential invariants. One class of the newly defined invariants is the object of a sharp isoperimetric inequality, while other new inequalities on known centro-affine invariants are obtained as a byproduct of the flow's study. Furt…

The paper classifies 3D hypersurfaces with specific geometric properties.

Comparison theorems in centro-affine differential geometry

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of …

New insights from centro-affine geometry solve a key geometric conjecture.

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses.

Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitab…

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

The paper connects curves on a light cone to KdV equations.

The paper explores conditions for statistical manifolds to be immersed and dual to each other.

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …

Study on the limits of projective special real manifolds and their symmetries.

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…

In this paper, we first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As an application in Finsler geometry, we gives some new characterizations of Berwald spaces.

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

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only…

We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving $p$-flow converges sequentially to the unit ball in the $…

I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied class of surfaces (n = 3). I extend an example of Akivis-Konnov, and use polynomial …

Sharp inequality for eigenvalues of convex bodies, proving ellipsoid uniqueness.

It is shown that the volume entropy of a Hilbert geometry associated to an $n$-dimensional convex body of class $C^{1,1}$ equals $n-1$. To achieve this result, a new projective invariant of convex bodies, similar to the centro-affine area, is constructed. In the case $n=2$, and without any assumption on the boundary, i…

A control system $\dot{q} = f(q,u)$ is said to be trivializable if there exists local coordinates in which the system is feedback equivalent to a control system of the form $\dot{q} = f(u)$. In this paper we characterize trivializable control systems and control systems for which, up to a feedback transformation, $f$ a…

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

Integrable flows on null curves in anti-de Sitter 3-space studied.

R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality …

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

The Funk metric connects billiards, projective geometry, and convex geometry.

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, nota…