A new duality transform (Q957553)

This is a review of recent results by the authors (without proofs). The paper deals with an abstract duality on a space of real-valued functions. This duality is described by an order reversing involution [cf. the authors, Electron. Res. Announc. Math. Sci. 14, 42--59 (2007; Zbl 1140.52300)]. The standard model of such duality for convex, lower-semi-continuous function \(f:\mathbb R^n\to [0,+\infty)\) gives the Lagrange transform \(L\) [cf. the authors, Ann. Math. (2) 169, No.~2, 661--674 (2009; Zbl 1173.26008)]. The authors describe an additional duality \(A\), which coincides with \(L\) for 2-homogeneous functions [cf. \textit{E. Milman}, Integral Equations Oper. Theory 57, No.~2, 217--228 (2007; Zbl 1139.46008)]. Both dualities are unique up to linear terms. Since \(AL = LA\), such composition is an order preserving involution different from the identity transformation. Similar results can be obtained for transformations in the family of convex subsets of \(\mathbb{R}^n\), which gives new understanding of support and Minkowski functionals [cf. the authors, J. Funct. Anal. 254, No.~10, 2648--2666 (2008; Zbl 1145.26003)].