Related papers: Affine function valued valuations

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…

In this paper, we study the $L_p$-Gaussian Minkowski problem, which arises in the $L_p$-Brunn-Minkowski theory in Gaussian probability space. We use Aleksandrov's variational method with Lagrange multipliers to prove the existence of the…

The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in $\mathbb{R}^n$ not necessarily symmetric with respect to the…

In this paper, we investigate an $L_p$ dual Christoffel-Minkowski type problem for the Hessian quotient operator $\frac{\sigma_{k}(\Lambda)}{\sigma_{l}(\Lambda)}$, where the operator $\Lambda$ has been widely studied in the literature.…

We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is…

We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation…

We study the convexity properties of the generalized trigonometric functions considered as functions of parameter. We show that $p\to\sin_p(y)$ and $p\to\cos_p(y)$ are log-concave on the appropriate intervals while $p\to\tan_p(y)$ is…

In this work, we will present variants Fixed Point Theorem for the affine and classical contexts, as a consequence of general Brouwer's Fixed Point Theorem. For instance, the affine results will allow working on affine balls, which are…

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on functions and measures. Alesker showed that these operations agree with the…

A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically…

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable…

We introduce a generalized version of Orlicz premia, based on possibly non-convex loss functions. We show that this generalized definition covers a variety of relevant examples, such as the geometric mean and the expectiles, while at the…

In this paper, we consider a class of Hessian equations associated to the $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$. By combining the tools of constant rank theorem, the a priori estimates and the continuity method, we…

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are…

For function spaces equipped with Muckenhoupt weights, the validity of continuous Sobolev embeddings in case $p_0\leq p_1$ is characterized. Extensions to Jawerth-Franke embeddings, vector-valued spaces and examples involving some prominent…

Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical…

The Shapley-Folkman theorem is a statement about the Minkowski sum of (non-convex) sets, expressing the closeness of the Minkowski sum to convexity in a quantitative manner. This paper establishes similar theorems for integrally convex…

We give estimates of the $L^p$ norm of the Bergman projection on a strongly pseudoconvex domain in $\mathbb{C}^n$. We show that this norm is comparable to $\frac{p^2}{p - 1}$ for $1 <p< \infty$.

The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant…

This article introduces $L^p$ versions of the support function of a convex body $K$ and associates to these canonical $L^p$-polar bodies $K^{\circ, p}$ and Mahler volumes $\mathcal{M}_p(K)$. Classical polarity is then seen as…