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Tropical convexity in location problems
We investigate location problems where the optimal solution is found within the tropical convex hull of the given input points. Our initial focus is...
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On Definitions of Finsler Spaces and Axiomatics of Singular Finsler Geometry
A review of various approaches to the concept of a Finsler space based on different definitions is given. In particular, the axiomatics of a singular...
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Firm non-expansive map**s in weak metric spaces
We introduce the notion of a firm non-expansive map** in weak metric spaces, extending previous work for Banach spaces and certain geodesic spaces....
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Affine convex geometry – Part 2
The main objective of this chapter is to study some important (starlike or convex) bodies associated to a given convex body. These are the so-called...
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The Geometry of the Thurston Metric: A Survey
This chapter is a survey about the Thurston metric on the Teichmüller space. The central issue is the construction of extremal Lipschitz maps between...
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On the bicompletion of a partial quasi-metric space and \(T_{0}\)-quasi-metric spaces
The purpose of the paper is to extend the completion theory of a partial metric space to the context of an asymmetric setup, namely, a partial...
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Computing Optimal Distances to Pareto Sets of Multi-Objective Optimization Problems in Asymmetric Normed Lattices
Given a finite dimensional asymmetric normed lattice, we provide explicit formulae for the optimization of the associated (non-Hausdorff) asymmetric...
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Symmetry of the B-J Orthogonality
While working with the usual orthogonality relation in Hilbert spaces, little do we realize that the symmetry of the orthogonality is a precious...
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Properties of the Distance Function to Strongly and Weakly Convex Sets in a Nonsymmetrical Space
We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and...
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Minimax Linear Estimation with the Probability Criterion under Unimodal Noise and Bounded Parameters
We consider a linear regression model with a vector of bounded parameters and a centered noise vector that has an uncertain unimodal distribution but...
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The Continuous Representation Property in Utility Theory
A topological space \((X,\tau )\) satisfies the Continuous...
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Historical steps of development of convexity as a field
In this chapter we will show historical steps of the development of convexity as a field and, in addition, developments of the relations between...
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Introduction
This introductory chapter contains a general description of the subject matter and a detailed outline of the content of the book.
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Classical inequalities
In this chapter, we collect some classical inequalities for geometric quantities of convex bodies (such as volume and surface area, for example).
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Introduction
Whenever an operation on a family of convex functions preserves convexity, one naturally wonders whether the subdifferential of this new function can...
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Noncommutative martingale Hardy-Orlicz spaces: Dualities and inequalities
We investigate dualities and inequalities related to noncommutative martingale Hardy-Orlicz spaces. More precisely, for a concave Orlicz function Φ,...
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Convex sets
Convexity is a very intuitive and geometric notion, but plays a fundamental role in many (also abstract) branches of mathematics. In vector spaces,...
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On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions
Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as...
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Eigenvalues and Eigenvectors
This chapter deals with the matrix eigenvalue problem, another major theme in linear algebra as important as the theory of linear equations. This...