Certificates of convexity for basic semi-algebraic sets

TL;DR

This paper introduces two numerical certificates to verify the convexity of basic semi-algebraic sets in real space, one exact and one sufficient, both derived from semidefinite programming solutions.

Contribution

It presents novel certificates for convexity of semi-algebraic sets, applicable via semidefinite programming, enhancing computational verification methods.

Findings

Certificates are derived from feasible solutions of semidefinite programs.

One certificate provides a necessary and sufficient condition.

The other offers a simpler sufficient condition.

Abstract

We provide two certificates of convexity for arbitrary basic semi-algebraic sets of $\R^n$. The first one is based on a necessary and sufficient condition whereas the second one is based on a sufficient (but simpler) condition only. Both certificates are obtained from any feasible solution of a related semidefinite program and so can be obtained numerically (however, up to machine precision).