Helly-type theorems for infinite and for finite intersections of sets starshaped via staircase paths.
Our main intention in this paper is to demonstrate how some seemingly purely geometric notions can be presented and understood in an analytic language of inequalities and then, with this understanding, can be defined for classes of functions and reveal new and hidden structures in these classes. One main example which we discovered is a new duality transform for convex non-negative functions on attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together...
A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic -space which, in the projective model for , is just the intersection of with a projective polyhedron whose vertices are all outside and whose edges all meet . We classify hyperideal polyhedra, up to isometries of , in terms of their combinatorial type and of their dihedral angles.
We show that the affine surface area as(∂K) of a convex body K in can be computed as where is a constant and is the illumination body.