相关论文: From valuations on convex bodies to convex functio…
This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus,…
In this study, multivalued generalizations of certain classes of single-valued transformations defined on metric spaces are obtained. Building upon recently introduced concepts such as mappings contracting perimeters of triangles, new…
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,…
The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area…
We investigate functionals defined on manifolds through parameterizations. If they are to be meaningful, from a geometrical viewpoint, they ought to be invariant under reparameterizations. Standard, local, integral functionals with this…
We study the local and global versions of the convexity, which is closely related to the problem of extending a convex function on a non-convex domain to a convex function on the convex hull of the domain and beyond the convex hull. We also…
In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the…
Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph…
A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the…
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…
As a generalization of geodesic function, this paper introduces the notion of geodesic $ \varphi_{E} $-convex function. Some properties of $ \varphi_{E} $-convex function and geodesic $ \varphi_{E} $-convex function are established. The…
We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…
We introduce a class of functional analogs of the symmetric difference metric on the space of coercive convex functions on $\mathbb{R}^n$ with full-dimensional domain. We show that convergence with respect to these metrics is equivalent to…
This paper reviews some recent applications of the theory of the compensated convex transforms or of the proximity hull as developed by the authors to image processing and shape interrogation with special attention given to the Hausdorff…
Inspired by the concept of hyperconvexity and its relation to curvature, we translate geometric properties of a metric space encoded by the curvature inequalities into the persistent homology induced by the \v{C}ech filtration of that…
McMullen's g-vector is important for simple convex polytopes. This paper postulates axioms for its extension to general convex polytopes. It also conjectures that, for each dimension d, a stated finite calculation gives the formula for the…
The classification of continuous, translation invariant Minkowski valuations which are contravariant (or covariant) with respect to the complex special linear group is established in a 2-dimensional complex vector space. Every such…