Related papers: Formal Adventures in Convex and Conical Spaces
The concept of a visible point of a convex set relative to a given point is introduced. A number of basic properties of such visible point sets is developed. In particular, it is shown that this concept is useful in the study of best…
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the…
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 note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the…
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…
This paper studies the configuration spaces of linkages whose underlying graph is a single cycle. Assume that the edge lengths are such that there are no configurations in which all the edges lie along a line. The main results are that,…
In this manuscript the idea of soft convex structures is given and some of their properties are investigated. Also, soft convex sets, soft concave sets and soft convex hull operator are defined and their properties are studied. Moreover,…
In this article we utilise abstract convexity theory in order to unify and generalize many different concepts from nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract…
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal…
We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it…
It is known that a real function $f$ is convex if and only if the set $$\mathrm{E}(f)=\{(x,y)\in\mathbb{R}\times\mathbb{R};\ f(x)\leq y\},$$ the epigraph of $f$ is a convex set in $\mathbb{R}^2$. We state an extension of this result for…
Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable…
The class of convex sets that admit approximations as Minkowski sum of a compact convex set and a closed convex cone in the Hausdorff distance is introduced. These sets are called approximately Motzkin-decomposable and generalize the notion…
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…
In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the…
In this paper we derive strong linear inequalities for sets of the form {(x, q) \in Rd \times R : q \geq Q(x), x \in Rd - int(P)}, where Q(x) : Rd \rightarrow R is a quadratic function, P \subset Rd and "int" denotes interior. Of particular…
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…
The cosine transforms of functions on the unit sphere play an important role in convex geometry, the Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting…
Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…