Related papers: Sharp geometric inequalities for the general $p$-a…

We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface…

Sharp affine fractional $L^p$ Sobolev inequalities for functions on $\mathbb R^n$ are established. The new inequalities are stronger than (and directly imply) the sharp fractional $L^p$ Sobolev inequalities. They are fractional versions of…

We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical…

For $p\in (-\infty,0)\cup(0,1)$ and a convex body $K\subset\mathbb{R}^n$ with the origin in its interior, we construct the family of $p$-affine dual curvature measures $\mathcal{I}_p(K,\cdot)$ with respect to $K$. The affine-invariant…

We establish sharp estimates for the $p$-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality. These estimates play an essential role in the study of…

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric…

In this paper, we define weighted relative $p(.)$-capacity and discuss properties of capacity in the space $W_{\vartheta }^{1,p(.)}(\mathbb{R}^{n}).$ Also, we investigate some properties of weighted variable Sobolev capacity. It is shown…

We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is…

We introduce an (equi-)affine invariant diffusion geometry by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to construct an invariant…

This paper presents a geometric approach to the classical isoperimetric problem by analysing the efficiency of regular polygons in enclosing maximum area for a fixed perimeter. Using efficiency metrics, it proves that regular polygons…

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 $L_p$…

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine…

We prove sharp inequalities for the average number of affine diameters through the points of a convex body $K$ in ${\mathbb R}^n$. These inequalities hold if $K$ is either a polytope or of dimension two. An example shows that the proof…

I extend the framework of rigid analytic geometry to the setting of algebraic geometry relative to monoids, and study the associated notions of separated, proper, and overconvergent morphisms. The category of affine manifolds embeds as a…

In this paper, we develop a refined analysis of hypergeometric functions to establish sharp quantitative integral inequalities for a general family of conformally invariant extension operators and their adjoints. Our results extend the…

Given a bounded open subset $\Omega$ of $\mathbb R^n$, we establish the weak closure of the affine ball $B^{\mathcal A}_p(\Omega) = \{f \in W^{1,p}_0(\Omega):\ \mathcal E_p f \leq 1\}$ with respect to the affine functional $\mathcal E_pf$…

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry,…

Symmetric Positive Definite (SPD) matrices have been widely used in medical data analysis and a number of different Riemannian met-rics were proposed to compute with them. However, there are very few methodological principles guiding the…

It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which…

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality…