Valuations on Convex Functions
All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.
math.FA↗
arXiv subjects
Andrea Colesanti
Publications and source records attributed to Andrea Colesanti.
Minkowski valuations under volume constraints
We provide a description of the space of continuous and translation invariant Minkowski valuations $Φ:\mathcal{K}^n\to\mathcal{K}^n$ for which there is an upper and a lower bound for the volume of $Φ(K)$ in terms of the volume of the convex body $K$ itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over $(n-1)$-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.
math.MG↗
Minkowski additive operators under volume constraints
We investigate Minkowski additive, continuous, and translation invariant operators $Φ:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $Φ(K)$ is bounded from above and below by multiples of the volume of the convex body $K$, uniformly in $K$. We obtain a representation result for an infinite subcone contained in the cone formed by this type of operators. Under the additional assumption of monotonicity or $SO(n)$-equivariance, we obtain new characterization results for the difference body operator.
math.MG↗
Monotonicity and concavity of integral functionals involving area measures of convex bodies
For a broad class of integral functionals defined on the space of $n$-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn-Minkowski type inequality. In particular, we prove that a Brunn-Minkowski type inequality implies monotonicity, and that a general Brunn-Minkowski type inequality is equivalent to the functional being a mixed volume.
math.MG↗
Valuations on the space of quasi-concave functions
We characterize the valuations on the space of quasi-concave functions defined on the $N$-dimensional Euclidean space, that are rigid motion invariant and continuous with respect to a suitable topology. Among them we also provide a specific description of those which are additionally monotone.
math.MG↗
On a linear refinement of the Prékopa-Leindler inequality
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are non-negative measurable functions, then the Prékopa-Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater or equal than the $0$-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa-Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.
math.FA↗
Quermassintegrals of quasi-concave functions and generalized Prékopa-Leindler inequalities
We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by $α$-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then, we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa-Leindler and Brascamp-Lieb inequalities. Further issues that we transpose to this functional setting are: integral-geometric formulae of Cauchy-Kubota type, valuation property and isoperimetric/Uryshon-like inequalities.
math.MG↗
A characterization of some mixed volumes via the Brunn-Minkowski inequality
We consider a functional $\mathcal F$ on the space of convex bodies in $\R^n$ defined as follows: ${\mathcal F}(K)$ is the integral over the unit sphere of a fixed continuous functions $f$ with respect to the area measure of the convex body $K$. We prove that if $\mathcal F$ satisfies an inequality of Brunn--Minkowski type, then $f$ is the support function of a convex body, i.e., $\mathcal F$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree $n-1$ and satisfy a Brunn--Minkowski type inequality.
math.MG↗
The area measure of log-concave functions and related inequalities
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.
math.FA↗
Isoperimetry and Stability of Hyperplanes for Product Probability Measures
We investigate stationarity and stability of half-spaces as isoperimetric sets for product probability measures, considering the cases of coordinate and non-coordinate half-spaces. Moreover, we present several examples to which our results can be applied, with a particular emphasis on the logistic measure.
math.FA↗
Functional inequalities derived from the Brunn-Minkowski inequalities for quermassintegrals
We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincaré type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space.
math.FA↗
On geometric properties of sets of positive reach in ${\bf E}^d$
Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\bf E}^d$ are proved. For $x_1$ and $x_2$ in ${\bf E}^d$ and $R>0$ let us denote by ${\mathfrak H}(x_1,x_2,R)$ the intersection of all closed balls of radius $R$ containing $x_1$ and $x_2$. For a compact subset $K$ of ${bf E}^d$ we prove that ${\rm reach}(K)\ge R$ if and only if for every $x_1,x_2\in K$ such that $\Vert x_1-x_2\Vert< 2R$, ${\mathfrak H}(x_1,x_2,R)\cap K$ is connected. A corollary is that if ${\rm reach}(K)\ge R>0$ and $D$ is a closed ball of radius less than or equal to $R$ (intersecting $K$) then ${\rm reach}(K\cap D)\ge R$. We also give a necessary and sufficient condition such that $A\subset{\bf E}^d$ admits a minimal cover (with respect to inclusion) of ${\rm reach}\ge R$.
math.MG↗
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 convex bodies with smooth boundary and positive Gauss curvature
math.FA↗
A sharp Rogers-Shephard type inequality for the p-difference body of a planar convex body
We prove a sharp Rogers-Shephard type inequality for the p-difference body of a convex body in the two-dimensional case, for every p greater than or equal to one.
math.MG↗
Functional inequalities related to the Rogers-Shephard inequality
For a real-valued non-negative and log-concave function we introduce a notion of difference function; the difference function represents a functional analog on the difference body of a convex body. We prove a sharp inequality which bounds the integral of the difference function from above in terms of the integral of the function itself, and we characterize equality conditions. The investigation is extended to an analogous notion of difference function for $α$-concave functions, with $α$ smaller than zero. In this case also, we prove an upper bound for the integral of the $α$-difference function of a function in terms of the integral of the function itself; the bound is proved to be sharp when $α$ is equal to minus infinity and in the one dimensional case.
math.MG↗