The first variation of the total mass of log-concave functions and related inequalities

Abstract

On the class of log-concave functions on R n , endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.2010MSC : 26B25 (primary), 26D10, 52A20.

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

“…if f is continuously differentiable on the whole R n (which also follows from (3.3) in case I(f ρ ) < +∞, for some ρ > 0). We point out that (3.5) may be seen as a variant of the integral representation formula given by Theorem 4.6 in [15]: in fact, (3.5) can be derived "formally" by applying Theorem 4.6 in [15] beyond its assumptions (more precisely, by taking therein ψ(y) = |y|).…”

“…whenever this limit exists. It was proved in [15] that, under suitable assumption of smoothness, decay at infinity and strict convexity of f (see Theorem 4.5 in [15] for the precise statement), the following representation formula holds:…”

“…The above definition is well-posed, since the mappings t → W i {f ≥ t} are monotone increasing, as a consequence of the monotonicity of the functionals W i (·) with respect to set inclusion. Actually, one can adopt the same natural extension from sets to functions in more general situations: If Φ is any functional with values in [0, +∞), defined on K n (or on the larger class of all Borel measurable subsets of R n ), and if it is monotone increasing with respect to set inclusion, one can extend it to the class Q n (respectively, to the class of all non-negative Borel measurable functions), by setting which are valid up to normalization constants for every K ∈ K n , the following definitions have been considered in the recent works [15,20,30,31], dealing especially with log-concave functions:…”

Rogers–Shephard inequality for log-concave functions

“…The notion of quermaßintegrals has also been extended from convex bodies to the setting of log-concave functions. In [18], [22] and [30], the case of the perimeter and the mean width is considered while, in [13], a different definition is given for all the quermaßintegrals. We will work with the definition in the latter paper, where, in particular, the quermaßintegral W 1 (surface area) of a log-concave function is defined by where c n = |B n 2 | |B n−1 2 | is a constant depending only on n and A n,1 is the set of affine 1-dimensional subspaces of R n and µ n,1 is the Haar probability measure on it.…”

Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

“…In 2013, Colesanti and Fragalà [35] introduced the "Minkowski addition" and "scalar multiplication," α • f ⊕ β • g (where α, β > 0), of log-concave functions f and g as…”