Linearization functors on real convex sets

We prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets. 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. We discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation.