Functional Analysis/Geometry
A new duality transform
Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1143-1148.

Continuing our search for dualities in different classes of functions, which usually turn out to have an essentially unique form, depending on the class, we exhibit a natural class of functions for which there are exactly two different types of duality transforms. One is the well known Legendre transform, and the other is new. We study the new transform, give a simple geometric interpretation for it, and present some applications.

Dans le cadre de notre étude de la dualité pour différentes classes de fonctions, souvent déterminée d'une façon unique par la classe, on exhibe une classe naturelle pour laquelle il y a exactement deux types de transformés de dualité. Une est la transformée de Legendre, et l'autre est nouvelle. Ces deux transformées ont des interprétations géométriques simples. On donne plusieurs applications des résultats.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.09.031
Artstein-Avidan, Shiri 1; Milman, Vitali 1

1 School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
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Artstein-Avidan, Shiri; Milman, Vitali. A new duality transform. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1143-1148. doi : 10.1016/j.crma.2008.09.031. http://www.numdam.org/articles/10.1016/j.crma.2008.09.031/

[1] Artstein-Avidan, S.; Klartag, B.; Milman, V. The Santaló point of a function, and a functional form of Santaló inequality, Mathematika, Volume 54 (2004), pp. 33-48

[2] Artstein-Avidan, S.; Milman, V. A characterization of the concept of duality, Electron. Res. Anounc. Math. Sci. AIMS, Volume 14 (2007), pp. 48-65

[3] S. Artstein-Avidan, V. Milman, The concept of duality in asymptotic geometric analysis, and the characterization of the Legendre transform, Ann. of Math., in press

[4] Artstein-Avidan, S.; Milman, V. The concept of duality for measure projections of convex bodies, J. Funct. Anal., Volume 254 (2007) no. 10, pp. 2648-2666

[5] Böröczky, K.; Schneider, R. A characterization of the duality mapping for convex bodies, Geom. Funct. Anal., Volume 18 (2008) no. 3

[6] Gruber, P. The endomorphisms of the lattice of norms in finite dimensions, Abh. Math. Sem. Univ. Hamburg, Volume 62 (1992), pp. 179-189

[7] Milman, V. Geometrization of probability (Kapranov, M.; Kolyada, S.; Manin, Y.; Moree, P.; Potyagailo, L., eds.), Geometry and Dynamics of Groups and Spaces, Progress in Mathematics, vol. 265, Birkhäuser, 2007, pp. 569-588

[8] R. Schneider, The endomorphisms of the lattice of closed convex cones, Beitraege Algebra Geom., in press

Cited by Sources:

Both authors were supported in part by the Israel Science Foundation: the first named author by grant No. 865/07, the second named author by grant No. 491/04. The authors were also supported in part by BSF grant No. 2006079.