Andrea Colesanti

FROM THE BRUNN–MINKOWSKI INEQUALITY TO A CLASS OF POINCARÉ-TYPE INEQUALITIES

We present an argument which leads from the Brunn–Minkowski inequality to a Poincare-type inequality on the boundary of a convex body K of class in Rn. We prove that for every ψ ∈ C1(∂K) Here denotes the (n - 1)-dimensional Hausdorff measure, νK is the Gauss map of K and DνK is the differential of νK i.e. the Weingarten map.

Brunn-Minkowski inequalities for two functionals involving the p -Laplace operator

In the family of n-dimensional convex bodies, we prove a Brunn-Minkowski type inequality for the first eigenvalue of the p-Laplace operator, or Poincaré constant, and for a functional extending the notion of torsional rigidity. In the latter case we also characterize equality conditions.

Steiner type formulae and weighted measures of singularities for semi-convex functions

For a given convex (semi-convex) function u, defined on a nonempty open convex set Ω ⊂ Rn, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for r ∈ {0, . . . , n}, the r-th coefficient measure of the local Steiner formula for u, restricted to the set of r-singular points of u, is absolutely continuous with respect to the r-dimensional Hausdorff measure, and that its density is the (n − r)-dimensional Hausdorff measure of the subgradient of u. As an application, under the assumptions that u is convex and Lipschitz, and Ω is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of r-singular points of u. Such estimates depend on the Lipschitz constant of u and on the quermassintegrals of the topological closure of Ω.

A Steiner type formula for convex functions

Given a convex function u , defined in an open bounded convex subset Ω of ℝ n , we consider the set where η is a Borel subset of Ω,ρ is nonnegative, and ∂ u ( x ) denotes the subgradient (or subdifferential) of u at x . We prove that P p ( u ; η) is a Borel set and its n -dimensional measure is a polynomial of degree n with respect to ρ. The coefficients of this polynomial are nonnegative measures defined on the Borel subsets of Ω. We find an upper bound for the values attained by these measures on the sublevel sets of u. Such a bound depends on the quermassintegrals of the sublevel set and on the Lipschitz constant of u. Finally we prove that one of these measures coincides with the Lebesgue measure of the image under the subgradient map of u .

A SHARP ROGERS AND SHEPHARD INEQUALITY FOR THE p-DIFFERENCE BODY OF PLANAR CONVEX BODIES

We prove a sharp Rogers and Shephard type inequality for the p-difference body of a convex body in the two-dimensional case, for every p ≥ 1.

Monotone Valuations on the Space of Convex Functions

Abstract We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.

Functional Inequalities related to the Rogers-Shephard Inequality

In this paper a notion of difference function Δ f is introduced for real-valued, non-negative and log-concave functions f defined in R n . The difference function represents a functional analogue of the difference body K + (− K ) of a convex body K. The main result is a sharp inequality which bounds the integral of Δ f from above in terms of the integral of f . Equality conditions are characterized. The investigation is extended to an analogous notion of difference function for α -concave functions, with α α -difference function of f in terms of the integral of f is proved. The bound is sharp in the case α = −∞ and in the one-dimensional case.

On the second differentiability of convex surfaces

Properties of pointwise second differentiability of real-valued convex functions in ℝn are studied. Some proofs of the Busemann-Feller-Aleksandrov theorem are reviewed and a new proof of this theorem is presented.

Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in ℝn with prescribed projection (n − 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.Conditions for the existence of bodies with minimumn-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.

Hessian measures of convex functions and applications to area measures

The Hessian measures of a (semi-)convex function can be introduced as coefficients of a local Steiner formula. The investigation of Hessian measures is continued by the provision of a geometric characterization of the support of these measures. Then the Radon-Nikodym derivative and the absolute continuity of Hessian measures with respect to Lebesgue measure are explored. As special cases of the results, known results for surface area measures of convex bodies are recovered.