Related papers: The Legendre transform, the Laplace transform and …
Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on…
A comparative analysis of two different versions of the Legendre transformation is presented. We provide an almost complete although somewhat superficial review of the geometric background for analytical mechanics. Complete coordinate…
Translation-invariant valuations on the space $L^\infty(\mathbb{R}^n)$ are examined. We prove that such functionals vanish on functions with compact support. Moreover a rich family of non-trivial translation-invariant valuations on…
A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally…
This paper provides an unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set valued functions.
This paper focuses on computing the convex conjugate (also known as the Legendre-Fenchel conjugate or c-transform) that appears in Euclidean Wasserstein-2 optimal transport. This conjugation is considered difficult to compute and in…
An introduction to geometric valuation theory is given. The focus is on classification results for $\operatorname{SL}(n)$ invariant and rigid motion invariant valuations on convex bodies and on convex functions.
A complete classification is established for continuous and SL(n) covariant matrix-valued valuations on Lp(Rn,|x|2dx). The assumption of matrix symmetry is eliminated. For n>2, such valuation is uniquely characterized by the moment matrix…
The task of finding a consistent relationship between a quantum Hamiltonian and a classical Lagrangian is of utmost importance for basic, but ubiquitous techniques like canonical quantization and path integrals. Nonconvex kinetic energies…
This paper uses the convolution theorem of the Laplace transform to derive new inverse Laplace transforms for the product of two parabolic cylinder functions in which the arguments may have opposite sign. These transforms are subsequently…
We reveal the relation between the Legendre transform of convex functions and heat flow evolution, and how it applies to the functional Blaschke-Santalo inequality. We also describe local maximizers in this inequality.
Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these…
In this paper, we introduce and analyze multidimensional vector-valued Laplace transform of functions with values in sequentially complete locally convex spaces. A great number of our results seem to be new even for the functions with…
We construct valuations on the space of finite-valued convex functions using integration of differential forms over the differential cycle associated to a convex function. We describe the kernel of this procedure and show that the…
All continuous translation invariant complex-valued valuations on Lebesgue measurable functions are completely classified. And all continuous rotation invariant complex-valued valuations on spherical Lebesgue measurable functions are also…
We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform,…
Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by…
The functions studied in the paper are quaternion-valued functions of a quaternionic variable. It is show that the left slice regular functions and right slice regular functions are related by a particular involution. The relation between…
We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h*-vectors. We give a surprisingly simple characterization of combinatorially positive…
Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…