Geodesic diameter of sets defined by few quadratic equations and inequalities

TL;DR

This paper establishes a polynomial bound on the geodesic diameter of sets defined by few quadratic equations and inequalities within the unit ball, improving upon the exponential bounds known for general polynomial degrees.

Contribution

It introduces a new polynomial bound for the geodesic diameter of quadratic-defined sets, combining techniques from D'Acunto, Kurdyka, and Barvinok.

Findings

Polynomial bound on geodesic diameter for quadratic sets

Improved bounds compared to general polynomial degree cases

Methodology combining geometric and algebraic techniques

Abstract

We prove a bound for the geodesic diameter of a subset of the unit ball in $\mathbb{R}^n$ described by a fixed number of quadratic equations and inequalities, which is polynomial in $n$, whereas the known bound for general degree is exponential in $n$. Our proof uses methods borrowed from D'Acunto and Kurdyka (to deal with the geodesic diameter) and from Barvinok (to take advantage of the quadratic nature).