Related papers: Equivariant Valuations on Convex Functions

Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived,…

We define and characterize spaces of manifold-valued generalized functions and generalized vector bundle homomorphisms in the setting of the full diffeomorphism-invariant vector-valued Colombeau algebra. Furthermore, we establish point…

We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the…

Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability…

The covariogram g_K(x) of a convex body K \subseteq E^d is the function which associates to each x \in E^d the volume of the intersection of K with K+x. Matheron asked whether g_K determines K, up to translations and reflections in a point.…

Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost…

We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems,…

We discuss $(K,N)$-convexity and gradient flows for $(K,N)$-convex functionals on metric spaces, in the case of real $K$ and negative $N$. In this generality, it is necessary to consider functionals unbounded from below and/or above,…

The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous…

We describe various structures of algebraic nature on the space of continuous valuations on convex sets, their properties (like versions of Poincar\'e duality and hard Lefschetz theorem), and their relations and applications to integral…

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and…

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that functions of bounded variation (BV functions) can be approximated in the strict sense and pointwise uniformly by special…

We discuss when law-invariant convex functionals "collapse to the mean". More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine…

In this note functions that transform open segments of a linear space into open segments of another linear space are studied and characterized. Assuming that the range is non-collinear, it is proved that such a map can always be expressed…

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping…

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and…

In this note, we introduce equi-affine invariants by averaging over the space of tropical structures of fixed covolume. Applied to the tropical distance series, this construction produces a family of equi-affine invariant functions…

Equivariance is a nice property to have as it produces much more parameter efficient neural architectures and preserves the structure of the input through the feature mapping. Even though some combinations of transformations might never…

Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The…

We give an alternative proof of a recent result of Klartag on the existence of almost subgaussian linear functionals on convex bodies. If $K$ is a convex body in ${\mathbb R}^n$ with volume one and center of mass at the origin, there exists…