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 [Formula: see text] or the dual function [Formula: see text] play the role of the inverses “[Formula: see text]” and “[Formula: see text]”, 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 [Formula: see text] one has [Formula: see text] where [Formula: see text]. 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.

Continued fractions built from convex sets and convex functions

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

Characterizing addition of convex sets by polynomiality of volume and by the homothety operation

We study addition operations between convex sets. We show that, under a short list of natural assumptions, one has polynomiality of volume only for the Minkowski addition. We also give two other characterization theorems. For the first theorem we define the induced homothety of an addition operation, and characterize additions by this homothety. The second theorem characterizes all additions which satisfy a short list of natural conditions.

Semicontinuous functions and convex sets in C(K) spaces

The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.