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

Artstein-Avidan, Shiri, and Milman, Vitali. "Hidden structures in the class of convex functions and a new duality transform." Journal of the European Mathematical Society 013.4 (2011): 975-1004. <http://eudml.org/doc/277371>.

@article{Artstein2011,
abstract = {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 $\mathbb \{R\}^n$ attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together with the classical Legendre transform, are essentially the only existing duality relations on this class of functions. Using these dualities we show that the geometric constructions of support and Minkowski functional may be extended, in a unique way, to the class of geometric log-concave functions, revealing hidden geometric structures on this class of functions.},
author = {Artstein-Avidan, Shiri, Milman, Vitali},
journal = {Journal of the European Mathematical Society},
keywords = {duality; convex functions; convex functions; duality; convexity in },
language = {eng},
number = {4},
pages = {975-1004},
publisher = {European Mathematical Society Publishing House},
title = {Hidden structures in the class of convex functions and a new duality transform},
url = {http://eudml.org/doc/277371},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Artstein-Avidan, Shiri
AU - Milman, Vitali
TI - Hidden structures in the class of convex functions and a new duality transform
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 4
SP - 975
EP - 1004
AB - 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 $\mathbb {R}^n$ attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together with the classical Legendre transform, are essentially the only existing duality relations on this class of functions. Using these dualities we show that the geometric constructions of support and Minkowski functional may be extended, in a unique way, to the class of geometric log-concave functions, revealing hidden geometric structures on this class of functions.
LA - eng
KW - duality; convex functions; convex functions; duality; convexity in
UR - http://eudml.org/doc/277371
ER -