# A lower bound on the positive semidefinite rank of convex bodies

## TL;DR

This paper establishes a new lower bound on the positive semidefinite rank of convex bodies, linking it to the degree of polynomials vanishing on the boundary of their polars, and demonstrates the bound's tightness for certain spectrahedra.

## Contribution

It introduces a novel lower bound on positive semidefinite rank based on algebraic degree, improving previous bounds and connecting to algebraic geometry and semidefinite programming.

## Key findings

- Lower bound of rac12;;;;;;;; ; for convex bodies in terms of polynomial degree.
- The bound is tight up to a constant factor for random spectrahedra of suitable dimension.

## Abstract

The positive semidefinite rank of a convex body $C$ is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body $C$ is at least $\sqrt{\log d}$ where $d$ is the smallest degree of a polynomial that vanishes on the boundary of the polar of $C$. This improves on the existing bound which relies on results from quantifier elimination. The proof relies on the B\'ezout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.