New higher-order equiaffine invariants

Abstract

Abstract. We introduce new affine invariants for smooth convex bodies. Some sharp affine isoperimetric inequalities are established for the new invariants.

Two Volume Product Inequalities and Their Applications

“…Besides their intrinsic interest, convex floating bodies, respectively, illumination bodies, have been useful in convex geometry in a number of ways. These bodies provide geometric interpretations of affine surface area, they appear in volume estimates for approximations of convex bodies by polytopes, and, more importantly, they generalize the definition of affine surface area to arbitrary convex bodies consistent with the other existing generalizations, while they surface in other applications as well, see [2,3,20,21,27,29,30,32,33]. In what concerns the extension of affine surface area, recall that Blaschke's original definition in R 3 , extended by Leichtweiss to higher dimensions, is so that a convex body K ⊂ R n+1 with boundary of class C 2 has affine surface area…”

Centro-Affine Invariants for Smooth Convex Bodies

“…Compared with the rate of change of the volume of a convex body L whose boundary is deformed by a normal vector field with speed v (as a function of u), which is The isoperimetric inequality for Ω 2 , p, generalizes Proposition 2.2 in [48] and we believe that the approach we had there can be extended to define iteratively the newly introduced Ω k,p for arbitrary convex bodies via weighted floating bodies. Implicitly, that will imply that Ω k,p (P ) is either zero or infinite on polytopes.…”

On the homothety conjecture

“…Floating bodies appear in many contexts and have been widely studied (see e.g. [1,2,4,5,6,8,13,14,15,19,20,21,23,26]). The homothety conjecture is among the problems related to floating bodies that was open for a long time.…”