Compact Quantitative Theories of Convex Algebras

We introduce the concept of compact quantitative equational theory. A quantitative equational theory is defined to be compact if all its consequences are derivable by means of finite proofs. We prove that the theory of interpolative barycentric (also known as convex) quantitative algebras of Mardare et. al. is compact. This serves as a paradigmatic example, used to obtain other compact quantitative equational theories of convex algebras, each axiomatizing some distance on finitely supported probability distributions.