Non-linear Plank Problems and polynomial inequalities

TL;DR

This paper investigates lower bounds for polynomial products in finite-dimensional Banach spaces and applies these findings to the plank problem, offering improvements over previous results especially with many polynomials.

Contribution

It introduces new lower bounds for polynomial norms in Banach spaces and enhances existing results related to the plank problem for large sets of polynomials.

Findings

Improved lower bounds for polynomial product norms in Banach spaces.

Enhanced results for the plank problem with many polynomials.

Applications demonstrating the theoretical bounds in finite-dimensional settings.

Abstract

We study lower bounds for the norm of the product of polynomials and their applications to the so called \emph{plank problem.} We are particularly interested in polynomials on finite dimensional Banach spaces, in which case our results improve previous works when the number of polynomials is large.