arXiv subjects
Publications and source records attributed to Andrea Colesanti.
In this paper, we study new extensions of the functional Blaschke-Santaló inequalities, and explore applications of such new inequalities beyond the classical setting of the standard Gaussian measure.
In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As an application, we give another proof of a Brunn-Minkowski type inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and of the log-concavity of the related first eigenfunction (both results have been proved in [9], by different methods).
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator \[ -Δ_{p,γ}u=-\text{div}(|\nabla u|^{p-2} \nabla u)+(x,\nabla u)|\nabla u|^{p-2}, \] where $p>1$, in the class of bounded Lipschitz domains in $\mathbb{R}^n$. We also prove that any corresponding positive eigenfunction is log-concave if the domain is convex.
We prove that the first (nontrivial) Dirichlet eigenvalue of the Ornstein-Uhlenbeck operator $$ L(u)=Δu-\langle\nabla u,x\rangle\,, $$ as a function of the domain, is convex with respect to the Minkowski addition, and we characterize the equality cases in some classes of convex sets. We also prove that the corresponding (positive) eigenfunction is log-concave if the domain is convex.
We show that every continuous and dually translation invariant valuation on the space of Lipschitz functions on the unit sphere of $\mathbb{R}^n$, $n\ge2$, can be decomposed uniquely into a sum of homogeneous valuations of degree $0$, $1$ and $2$. In particular, there does not exist any non-trivial, continuous and dually translation invariant valuation which is homogeneous of degree $3$ or higher. For the space of those of degree $0$, $1$ and $2$ we provide a description of a dense subspace.
A new version of the Hadwiger theorem on convex functions is established and an explicit representation of functional intrinsic volumes is found using new functional Cauchy-Kubota formulas. In addition, connections between functional intrinsic volumes and their classical counterparts are obtained and non-negative valuations are classified.
A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb R}^n$ is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.
A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.
New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domains is proved, and its connection to the Abel transform is explained.
An Orlicz version of the $L_p$-Minkowski problem on $S^{n-1}$ is discussed corresponding to the case $-n<p<0$.
We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n-1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.
We provide an integral representation for continuous, rotation invariant and dot product invariant valuations defined on the space Lip$(S^{n-1})$ of Lipschitz continuous functions on the unit $n-$sphere.
Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang \cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors \cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequality for log-concave measures, which in the case of Gaussian measure was conjectured by Gardner and Zvavitch \cite{GZ}. In this note, we obtain stability results for both Log-Brunn-Minkowski and dimensional Brunn-Minkowski inequalities for rotation invariant log-conave measures near a ball. Remarkably, the assumption of symmetry is only necessary for Log-Brunn-Minkowski stability, which emphasizes an important difference between the two conjectured inequalities. Also, we determine the infinitesimal version of the log-Brunn-Minkowski inequality. As a consequence, we obtain a strong Poincaré-type inequality in the case of unconditional convex sets, as well as for symmetric convex sets on the plane. Additionally, we derive an infinitesimal equivalent version of the B-conjecture for an arbitrary measure.
We discuss the smoothness and strict convexity of the solution of the $L_p$ Minkowski problem when $p<1$ and the given measure has a positive density function.
Chou and Wang's existence result for the $L_p$-Minkowski problem on ${\mathbb S}^{n-1}$ for $p\in(-n,1)$ and an absolutely continuous measure $μ$ is discussed and extended to more general measures. In particular, we provide an almost optimal sufficient condition for the case $p\in(0,1)$.
We prove that the log-Brunn-Minkowski inequality \begin{equation*} |λK+_0 (1-λ)L|\geq |K|^λ|L|^{1-λ} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when $K\subset\mathbb{R}^n$ is a ball and $L$ is a symmetric convex body in a suitable $C^2$ neighborhood of $K$.
A classification of SL$(n)$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new SL$(n)$ covariant Minkowski valuation on convex functions is defined and characterized.
We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations.