Linearization functors on real convex sets

TL;DR

This paper introduces new functorial operations on real convex sets derived from linearizing polynomial optimization problems, enabling efficient computation and novel convex constructions analogous to classical algebraic functors.

Contribution

It develops convex analogues of algebraic functors like Hom, tensor, and Schur functors, expanding the toolkit for convex set analysis and optimization.

Findings

Operations can be computed or approximated efficiently.

New convex constructions for polyhedra are introduced.

Functorial properties extend classical algebraic concepts to convex sets.

Abstract

We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to efficient computation. These operations are convex analogues of Hom functors, tensor products, symmetric powers, exterior powers and general Schur functors on vector spaces and lead to novel constructions even for polyhedra.