Algebraically inspired results on convex functions and bodies

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body K° or the dual function φ∗ play the role of the inverses "K-1" and "φ-1", we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function φ one has (φ + δ)∗ + (φ ∗ + δ) ∗ = δ, where δ(x) = 1 2 x2. We also prove several corollaries of this identity, including a Santal type inequality and a contribution to the theory of summands. We proceed to discuss the analogous identity for convex bodies, where an unexpected distinction appears between the classical Minkowski addition and the more modern 2-addition. In the final section of the paper we consider the harmonic and geometric means of convex bodies and convex functions, and discuss their concavity properties. Once again, we find that in some problems the 2-addition of convex bodies behaves even better than the Minkowski addition.