On the nontrivial projection problem

TL;DR

This paper investigates the nontrivial projection problem in finite-dimensional normed spaces and convex bodies, demonstrating that the conjecture holds approximately up to a logarithmic factor.

Contribution

It provides a partial positive answer to the nontrivial projection problem, showing the existence of well-bounded projections up to a logarithmic factor.

Findings

The problem is true up to a logarithmic factor.

Every finite-dimensional normed space admits a well-bounded projection with certain bounds.

Centrally symmetric convex bodies are approximately affinely equivalent to a product of two bodies of non-trivial dimension.

Abstract

The Nontrivial Projection Problem asks whether every finite-dimensional normed space of dimension greater than one admits a well-bounded projection of non-trivial rank and corank or, equivalently, whether every centrally symmetric convex body (of arbitrary dimension greater than one) is approximately affinely equivalent to a direct product of two bodies of non-trivial dimension. We show that this is true "up to a logarithmic factor."