Seminar on Banach and Metric Space Geometry

Date: December 3, 2021

Time: 09:00AM - 10:00PM

Location: Zoom

Speaker: Kasia Wyczesany, Tel Aviv University

Title: On almost Euclidean and well-complemented subspaces of finite-dimensional normed spaces

Abstract: In this talk I will discuss a version of an old question of Vitali Milman about almost Euclidean and well-complemented subspaces. In particular, I will introduce a notion of ' ε-good points ', which allows for a convenient reformulation of the problem. Let (X,||·||_X) be a normed space. It turns out that if a linear subspace Y ⊂ X consists entirely of ε-good points then the restriction of the norm ||·||_X to Y must be approximately a multiple of the l₂ norm and the operator norm of the orthogonal projection onto Y is close to 1. I will present an example of a normed space X of arbitrarily high dimension, whose Banach-Mazur distance from the l₂^{dim X} is at most 2, but such that non of its (even two-dimensional) subspaces consists entirely of ε-good points. The talk is based on joint work with Timothy Gowers.