A sequence of inequalities for certain sets of concurrent cevians.
A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets
The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this...
A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.
A sharp isoperimetric inequality in the plane
We show that among all the convex bounded domain in having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.
A sharpening of a discrete analog of Wirtinger's and isoperimetric inequalities
A sharpening of a discrete case of Wirtinger's inequality is given. It is then used to sharpen the isoperimetric unequality for polygons.
A sharpening of discrete analogues of Wirtinger's inequality
A short introduction to shadows of 4-manifolds
We give a self-contained introduction to the theory of shadows as a tool to study smooth 3-manifolds and 4-manifolds. The goal of the present paper is twofold: on the one hand, it is intended to be a shortcut to a basic use of the theory of shadows, on the other hand it gives a sketchy overview of some of the most recent results on shadows. No original result is proved here and most of the details of the proofs are left out.
A short proof, based on mixed volumes, of Liggett's theorem on the convolution of ultra-logconcave sequences.
A short proof of Dvoretzky's theorem on almost spherical sections of convex bodies
A short proof of rigidity of convex polytopes.
A Simple and Relatively Efficient Triangulation of the n-Cube.
A simple geometrie proof of a theorem for starshaped unions of convex sets
A Simple Proof of the Kinematic Formula.
A simpler construction of volume polynomials for a polyhedron.
A Simpler Proof of the Negative Association Property for Absolute Values of Measures Tied to Generalized Orlicz Balls
Negative association for a family of random variables means that for any coordinatewise increasing functions f,g we have for any disjoint sets of indices (iₘ), (jₙ). It is a way to indicate the negative correlation in a family of random variables. It was first introduced in 1980s in statistics by Alem Saxena and Joag-Dev Proschan, and brought to convex geometry in 2005 by Wojtaszczyk Pilipczuk to prove the Central Limit Theorem for Orlicz balls. The paper gives a relatively simple proof of...
A stability estimate for the Aleksandrow-Fnechel inequality, with an application to mean curvature.
A Stability Property of the Densest Circle Packing.
A stability theorem for the area sum of a convex polygon and its polar reciprocal.
A staircase illumination theorem for orthogonal polygons.
A study of variational inequalities for set-valued mappings.