A Convex Form That Is Not a Sum of Squares

Explore a 38-minute lecture from the Fields Institute's Workshop on Real Algebraic Geometry and Algorithms for Geometric Constraint Systems, delivered by James Saunderson of Monash University. Delve into the intriguing world of convex forms and sum of squares, addressing Parrilo's question on whether all convex forms are sums of squares. Discover Blekherman's nonconstructive argument proving the existence of convex forms that are not sums of squares, and examine an explicit example of a degree-four convex form in 272 variables that defies this property. Investigate the form's connection to the Cauchy-Schwarz inequality over octonions and explore the implications for sum-of-squares-based relaxations in polynomial optimization problems over the sphere.