JournalsjemsVol. 13, No. 4pp. 975–1004

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

  • Vitali Milman

    Tel Aviv University, Israel
  • Shiri Artstein-Avidan

    Tel Aviv University, Israel
Hidden structures in the class of convex functions and a new duality transform cover
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Abstract

Our main intension 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 within these classes. One main example which we discovered is a new duality transform for convex non-negative functions on ℝ_n_ attaining thevalue 0 at the origin (which we called “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.

Cite this article

Vitali Milman, Shiri Artstein-Avidan, Hidden structures in the class of convex functions and a new duality transform. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 975–1004

DOI 10.4171/JEMS/273