arXiv research

This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine $L^{p}-$Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…

The paper develops inequalities for log-concave functions and related surface areas.

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-var…

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on $\mathbb{R}^n$. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which i…

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

This paper aims to develop basic theory for the dual Orlicz $L_φ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequalit…

Paper finds inequalities for convex domains in hyperbolic space.

Extends Choi-Wang inequality to Li-Xia affine connections.

A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.

We prove isoperimetric inequalities for quotients of $n$-dimensional Affine buildings. We use these inequalities to prove topological overlapping for the 2-dimensional skeletons of these buildings.

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

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

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…

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

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

The paper studies a new class of affine maximal surfaces with singularities.

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close en…

Develops a theory for mth order p-affine capacity for convex bodies containing the origin.

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

The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_φ$ affine and geominimal surface areas for single convex body as well as for multiple convex bod…

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $Γ_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ …

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…

Study intrinsic volume forms on complex hypersurfaces.

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…

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

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

Abstract mathematical formulas for statistical structures and curvatures.

The paper explores fully affine maximal curves and their properties.

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

New bounds for convex clustering under graph connectivity.

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…

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

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

The paper extends inequalities for projection bodies to arbitrary measures.

This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…

In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual a…

The study classifies certain types of incomplete surfaces with low curvature.

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, …

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…

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 the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.

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 …