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The Krein–Milman Theorem for Homogeneous Polynomials
This note addresses the problem of recovering a convex set of homogeneous polynomials from the subset of its extreme points, i.e., the justification...
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The separation of convex sets and the Krein–Milman theorem in fuzzy quasi-normed space
Motivated by some deep problems in optimization and control theory, convexity theory has been extended to the various infinite dimensional functional...
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The Krein–Milman Theorem
The Krein–Milman theorem asserts that in a Hausdorff locally convex space all points of a compact convex set can be approximated by convex...
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Archimedean Representation Theorem for modules over a commutative ring
Pólya’s Positivstellensatz and Handelman’s Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem...
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Cospanning Characterizations of Violator and Co-violator Spaces
Given a finite set E and an operator \(\sigma :2^{E}\longrightarrow 2^{E}\)...
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A geometric Jordan decomposition theorem
For a compact convex set K , let A ( K ) denote the space of real-valued affine continuous functions, equipped with the supremum norm. For a closed...
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The Krein–Milman Theorem and Hansen’s Variant of the Hansen–Pedersen Proof
In this chapter, we will present a proof of Loewner’s theorem due to Hansen–Pedersen that relies on the Krein–Milman theorem; we follow a variant of...
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Some basic definitions and tools
In this short chapter we shall provide some basic information on the theory of locally convex spaces and related topics that will be needed in the...
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Uniform distribution of sequences and its interplay with functional analysis
In this paper we apply ideas from the theory of Uniform Distribution of sequences to Functional Analysis and then drawing inspiration from the...
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Locally Convex Spaces
Recall that Theorems 2.6.4 and 2.6.7 established the HB...
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Dilation theory in finite dimensions and matrix convexity
We establish a finite-dimensional version of the Arveson-Stinespring dilation theorem for unital completely positive maps on operator systems. This...
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The Krein–Milman Theorem and Its Applications
One of the main merits of the functional analysis-based approach to problems of classical analysis is that it reduces problems formulated...
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On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces
In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the...
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A Course on Topological Vector Spaces
This book provides an introduction to the theory of topological vector spaces, with a focus on locally convex spaces. It discusses topologies in dual...
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On a question of supports
We give a sufficient condition in order that n closed connected subsets in the n -dimensional real projective space admit a common multitangent...
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Convexity on Convex Polyhedra
We have set as our goal proving that there is a vm-reductionvertex-merge reduction ordering of the set V of vertices to either side of a...
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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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Linear inverse problems with Hessian–Schatten total variation
In this paper, we characterize the class of extremal points of the unit ball of the Hessian–Schatten total variation (HTV) functional. The underlying...
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Introduction
The study and classification of the extreme points of the unit ball of a Banach space is a classical problem in functional analysis. This question is...