Valuations on Log-Concave Functions

A classification of SL(n) and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as SL(n) and translation invariant valuations on log-concave functions. 2000 AMS subject classification: 26B25 (46B20, 52A21, 52A41, 52B45) A function Z defined on a lattice (L,∨,∧) and taking values in an abelian semigroup is called a valuation if Z(f ∨ g) + Z(f ∧ g) = Z(f) + Z(g), (1) for all f, g ∈ L. A function Z defined on a set S ⊂ L is called a valuation if (1) holds whenever f, g, f ∨g, f ∧g ∈ S. In the classical theory, valuations on the set of convex bodies (non-empty, compact, convex sets), Kn, in Rn are studied, where ∨ and ∧ denote union and intersection, respectively. Valuations played a critical role in Dehn’s solution of Hilbert’s Third Problem and have been a central focus in convex geometric analysis. In many cases, well known functions in geometry could be characterized as valuations. For example, a first classification of the Euler characteristic and volume as continuous, SL(n) and translation invariant valuations on Kn was established by Blaschke [5]. The celebrated Hadwiger classification theorem [17] gives a complete classification of continuous, rotation and translation invariant valuations on Kn and provides a characterization of intrinsic volumes. Alesker [2] obtained classification results for translation invariant valuations. Since several important geometric operators like the Steiner point and the moment vector are not translation invariant, also translation covariance played an important role. In particular, Hadwiger & Schneider [18] characterized linear combinations of quermassvectors as continuous, rotation and translation covariant vector-valued valuations. In addition to the ongoing research on real-valued valuations on convex bodies [1, 14, 19, 22, 28, 29], valuations with values in Kn have attracted interest. Such a map is called a Minkowski valuation if the addition in (1) is given by Minkowski addition, that is K + L = {x+ y : x ∈ K, y ∈ L} for K,L ∈ Kn. The first results in this direction were established by Ludwig [23,24]. See [13,15,21,40,43] for some of the pertinent results. More recently, valuations were defined on function spaces. For a space S of real-valued functions we denote by f ∨ g the pointwise maximum of f and g while f ∧ g denotes their pointwise minimum. For Sobolev spaces [25,26,30] and Lp spaces [27,35,41,42] complete classifications of valuations intertwining with the SL(n) were established. For definable functions,