Α-concave Functions and a Functional Extension of Mixed Volumes

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation ⊕ on the class of quasi-concave functions, such that every class of α-concave functions is closed under ⊕. We then define the mixed integrals, which are the polarization of the integral with respect to ⊕. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-concave functions, this is done by restating those results in the language of rearrangement inequalities. Restricting ourselves to α-concave functions, we state a generalization of the Alexandrov inequalities in their more familiar form. 1. α-concave functions Let us begin by introducing our main objects of study: Definition 1. Fix −∞ ≤ α ≤ ∞. We say that a function f : R → [0,∞) is α-concave if f is supported on some convex set Ω, and for every x, y ∈ Ω and 0 ≤ λ ≤ 1 we have f (λx+ (1− λ)y) ≥ [λf(x) + (1− λ) f(y)] 1 α . For simplicity, we will always assume f is upper semicontinuous, maxx∈Rn f(x) = 1, and f(x)→ 0 as |x| → ∞. The class of all such α-concave functions will be denoted by Cα (R). In the above definition, we follow the convention set by Brascamp and Lieb ([6]), but the notion of α-concavity may be traced back to Avriel ([1]) and Borell ([4], [5]). Discussions of α-concave functions from a geometric point of view may be found, e.g., in [2] and [9]. In the cases α = −∞, 0,∞ we understand Definition 1 in the limit sense. For example, f ∈ C∞ (R) if f is supported on some convex set Ω, and f (λx+ (1− λ)y) ≥ max {f(x), f(y)} Received by the editors November 13, 2012 and, in revised form, December 16, 2012. 2010 Mathematics Subject Classification. 52A39, 26B25.