Extremal sections and projections of certain convex bodies: a survey

TL;DR

This survey reviews sharp estimates on the volumes of sections and projections of convex bodies, especially lex_p balls, highlighting probabilistic and analytic methods developed over decades in the geometry of numbers.

Contribution

It compiles and discusses recent advances and methods in estimating volumes of sections and projections of convex bodies, emphasizing lex_p balls.

Findings

Compilation of sharp volume estimates for convex body sections and projections.

Overview of probabilistic and analytic techniques used in the field.

Highlights of historical development in geometry of numbers.

Abstract

We survey results concerning sharp estimates on volumes of sections and projections of certain convex bodies, mainly $\ell_p$ balls, by and onto lower dimensional subspaces. This subject emerged from geometry of numbers several decades ago and since then has seen development of a variety of probabilistic and analytic methods, showcased in this survey.