Narrow Results

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.

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 $K^{\circ }$ or the dual function ${\phi }^{\ast }$ play the role of the inverses “$K^{-1}$” and “${\phi }^{-1}$”, 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 $\phi$ one has