Related papers: Some inequalities of isoperimetric type for the c-…
We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci…
In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $\Omega$, different from a ball, which minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the…
Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities,…
In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related…
We prove that, among all subsets $\Omega\subset \mathbb{C}$ having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\geq 1$) of the corresponding Toeplitz operator…
We prove the Blaschke-Santal\'o inequality restricted to $n$-gons: the extremal polygons are the affine regular $n$-gons. If either the John or the L\"owner ellipse of a planar $o$-symmetric convex body $K$ is the unit circle about $o$,…
Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to…
We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary…
Let \Sigma be a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of \Sigma is bounded from below by the volume of the unit ball in R^k. This answers a question posed by R.…
In this paper we study the following quantitative isoperimetric inequality in the plane: $\lambda_0^2(\Omega) \leq C \delta(\Omega)$ where $\delta$ is the isoperimetric deficit and $\lambda_0$ is the barycentric asymmetry. Our aim is to…
In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the…
For every given $\beta<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda_\beta(\Omega)$ among convex (not necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter. In particular,…
In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the…
Motivated by the Blaschke-Santal\'o inequality, we define for a convex body K in ${\bf R}^n$ and for $t \in {\bf R}$ the Santal\'o-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine…
We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in $\mathbb{C}^n$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic…
For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad {\rm os}_{\psi}(K):=\inf_{K^\prime\supset…
The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed…
Motivated by a long-standing conjecture of Polya and Szeg\"o about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the…
In 1921 Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that $#(K\cap\Z^n)\leq n! \vol(K)+n$, whenever $K\subset\R^n$ is a convex body…