Related papers: Isoperimetric inequalities in high-dimensional con…
We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that $$S(K)\leq\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in…
We compare and combine two approaches that have been recently introduced by Dafnis and Paouris [DP] and by Klartag and Milman [KM] with the aim of providing bounds for the isotropic constants of convex bodies. By defining a new hereditary…
In this paper we first study the isoperimetric problem in the case of integer triangles, as well as Alcuin's sequence and how it relates to the number of different integer triangles with a given perimeter. We then present and compare two…
The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification…
Problems in optimization and geometric probability are discussed, all connected with angles subtended at an observer's eye by an object at a distance. Several of these remain unsolved.
We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev…
This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and…
We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq…
We review our recent results on the problem of optimizing Riesz means of Laplace eigenvalues among convex sets of given measure in the regime where the cut-off parameter in the definition of the Riesz means tends to infinity. We show that…
This note reviews the studies of the last decades emphasizing a common principle based on entropy, logarithmic Sobolev inequality and hypercontractivity, behind four most celebrated inequalities by M. Talagrand: the convex distance…
We discuss an analytic form of the dilation inequality for symmetric convex sets in Euclidean spaces, which is a counterpart of analytic aspects of Cheeger's isoperimetric inequality. We show that the dilation inequality for symmetric…
We prove an inequality that extends to arbitrary measures the hyperplane inequality for volume of unconditional convex bodies originally observed by Bourgain.
We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically…
We deal with an isoperimetric problem on the Finsler hyperbolic plane. The space is defined as the Lie group of proper affine transformations of the line with a left-invariant Finsler structure. To state the problem, we use the…
We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems. Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a…
We derive a relative version of the slicing Bennequin inequalities for cobordant Legendrian knots, and review a few proofs of the result.
The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by…
A number of geometric inequalities for convex sets arising from Brunn's concavity principle have recently been shown to yield local stochastic formulations. Comparatively, there has been much less progress towards stochastic forms of…