A remark about the directed line-elements in the plane
A remark on mixed curvature measures for sets with positive reach.
A representation for volume involving interior reach.
A reverse isoperimetric inequality for convex plane curves.
A sandwich theorem and Hyers-Ulam stability of affine functions.
A second order -approximation method for constrained optimization problems involving second order invex functions
A new approach for obtaining the second order sufficient conditions for nonlinear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order -approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order -approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear...
A selection theorem of Helly type and its applications
We prove an abstract selection theorem for set-valued mappings with compact convex values in a normed space. Some special cases of this result as well as its applications to separation theory and Hyers-Ulam stability of affine functions are also given.
A separation theorem for finite families
A seperation property of plane convex sets.
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 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 simple geometrie proof of a theorem for starshaped unions of convex sets
A Simple Proof of the Kinematic Formula.
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...