The geometry of Euclidean convolution inequalities and entropy

Euclidean Geometry before non-Euclidean Geometry

In [3], in my argument for the primacy of Euclidean geometry on the basis of rigid motions and the existence of similar but non-congruent triangles, I wrote the following: A: “The mobility of rigid objects is now recognized as one of the things every normal human child learns in infancy, and this learning appears to be part of our biological progaramming.” B. “. . . we are all used to thinking ...

Forward and Reverse Entropy Power Inequalities in Convex Geometry

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the proce...

Links between the Logarithmic Sobolev Inequality and the convolution inequalities for Entropy and Fisher Information

Relative to the Gaussian measure on R, entropy and Fisher information are famously related via Gross’ logarithmic Sobolev inequality (LSI). These same functionals also separately satisfy convolution inequalities, as proved by Stam. We establish a dimension-free inequality that interpolates among these relations. Several interesting corollaries follow: (i) the deficit in the LSI satisfies a conv...

extensions of some polynomial inequalities to the polar derivative

توسیع تعدادی از نامساوی های چند جمله ای در مشتق قطبی

Basic Notions of Geometry and Euclidean Geometry