Spectrahedral shadows

TL;DR

This paper demonstrates that many convex semi-algebraic sets in Euclidean space cannot be represented as spectrahedral shadows, providing a negative answer to a longstanding open question and disproving the Helton-Nie conjecture.

Contribution

The paper proves the non-existence of semidefinite representations for many convex semi-algebraic sets, refuting the Helton-Nie conjecture.

Findings

Many convex semi-algebraic sets lack semidefinite representations

Disproof of the Helton-Nie conjecture

Negative answer to Nemirovski's question

Abstract

We show that there are many (compact) convex semi-algebraic sets in euclidean space that do not have a semidefinite representation. This gives a negative answer to a question by Nemirovski, resp. it shows that the Helton-Nie conjecture is false.