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Hidden structures in the class of convex functions and a new duality transform

Shiri Artstein-Avidan, Vitali Milman (2011)

Journal of the European Mathematical Society

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 n attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together...

Hyperideal polyhedra in hyperbolic 3-space

Xiliang Bao, Francis Bonahon (2002)

Bulletin de la Société Mathématique de France

A hyperideal polyhedron is a non-compact polyhedron in the hyperbolic 3 -space 3 which, in the projective model for 3 ℝℙ 3 , is just the intersection of 3 with a projective polyhedron whose vertices are all outside 3 and whose edges all meet 3 . We classify hyperideal polyhedra, up to isometries of 3 , in terms of their combinatorial type and of their dihedral angles.

Illumination bodies and affine surface area

Elisabeth Werner (1994)

Studia Mathematica

We show that the affine surface area as(∂K) of a convex body K in n can be computed as a s ( K ) = l i m δ 0 d n ( v o l n ( K δ ) - v o l n ( K ) ) / ( δ 2 / ( n + 1 ) ) where d n is a constant and K δ is the illumination body.

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