Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms

TL;DR

This paper explores key problems in asymptotic convex geometry and random matrices, inspired by the simplex method in linear programming, focusing on projections of high-dimensional polytopes and matrix norm estimates.

Contribution

It discusses conjectures and known results related to high-dimensional projections and random matrix norms, advancing understanding in these interconnected areas.

Findings

Analysis of projections of high-dimensional polytopes

Estimates of norms of random matrices and their inverses

Discussion of conjectures and known results in the field

Abstract

The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses.