4 citations · 6 across the 11 of their papers we have counts for
17 papers
Empirical dual volumes
Grigoris Paouris, Peter Pivovarov, Paul Simanjuntak
We introduce an empirical version of dual volumes within dual Brunn--Minkowski theory. These provide a simple model for computation and analysis under symmetrization, including mom…
Empirical forms of the Petty projection inequality
Grigoris Paouris, Peter Pivovarov, Kateryna Tatarko
The Petty projection inequality is a fundamental affine isoperimetric principle for convex sets. It has shaped several directions of research in convex geometry which forged new co…
From intersection bodies to dual centroid bodies: a stochastic approach to isoperimetry
Radoslaw Adamczak, Grigoris Paouris, Peter Pivovarov +1
We establish a family of isoperimetric inequalities for sets that interpolate between intersection bodies and dual Lp centroid bodies. This provides a bridge between the Busemann i…
Limit theorems for the volumes of small codimensional random sections of -balls
Radosław Adamczak, Peter Pivovarov, Paul Simanjuntak
We establish Central Limit Theorems for the volumes of intersections of (the unit ball of ) with uniform random subspaces of codimension for fixed and $…
Stochastic forms of Brunn's principle
P. Pivovarov, J. Rebollo Bueno
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 ha…
A stochastic Prekopa-Leindler inequality for log-concave functions
Peter Pivovarov, Jesus Rebollo Bueno
The Brunn-Minkowski and Prékopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for i…
Randomized Urysohn-type inequalities
Thomas Hack, Peter Pivovarov
As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic…
Remarks on the Rényi Entropy of a sum of IID random variables
Benjamin Jaye, Galyna V. Livshyts, Grigoris Paouris +1
In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the Rényi entropy of the sum of independent random varia…
Spherical centroid bodies
Florian Besau, Thomas Hack, Peter Pivovarov +1
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in…
Affine isoperimetric inequalities on flag manifolds
Susanna Dann, Grigoris Paouris, Peter Pivovarov
Building on work of Furstenberg and Tzkoni, we introduce -flag affine quermassintegrals and their dual versions. These quantities generalize affine and dual affine quermas…
On a quantitative reversal of Alexandrov's inequality
Grigoris Paouris, Peter Pivovarov, Petros Valettas
Alexandrov's inequalities imply that for any convex body , the sequence of intrinsic volumes is non-increasing (when suitably normalized). Milman's random…
Randomized isoperimetric inequalities
Grigoris Paouris, Peter Pivovarov
We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their…
Random ball-polyhedra and inequalities for intrinsic volumes
Grigoris Paouris, Peter Pivovarov
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedur…
On sharp bounds for marginal densities of product measures
Galyna Livshyts, Grigoris Paouris, Peter Pivovarov
We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if is a probability density on of the form $f(x)=\prod_{i=1…
Bounding marginal densities via affine isoperimetry
Susanna Dann, Grigoris Paouris, Peter Pivovarov
Let be a probability measure on with a bounded density . We prove that the marginals of on most subspaces are well-bounded. For product measures, studied…
Volume of the polar of random sets and shadow systems
Dario Cordero-Erausquin, Matthieu Fradelizi, Grigoris Paouris +1
We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers a…
A central limit theorem for projections of the cube
Grigoris Paouris, Peter Pivovarov, Joel Zinn
We prove a central limit theorem for the volume of projections of the N-cube onto a random subspace of dimension n, when n is fixed and N tends to infinity. Randomness in this case…