Topic
Polyhedral space
About: Polyhedral space is a research topic. Over the lifetime, 40 publications have been published within this topic receiving 320 citations.
Papers
TL;DR: In this article, it was shown that the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm, which is itself contained by the set NA of norm attaining functionals.
Abstract: In any dual space X, the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm which is itself contained in the set NA of norm attaining functionals. We show that NA and SSD coincide if and only if every proximinal hyperplane of X is strongly proximinal, and that if QP and NA coincide then every finite codimensional proximinal subspace of X is strongly proximinal. Natural examples and applications are provided.
67 citations
6 Jan 1988
TL;DR: It is proved that max flow equals min cut; the problem of constructing min cuts and max flows is addressed; and the continuous Dijkstra paradigm of solving shortest paths problems corresponds to a continuous version of Berge's algorithm for computation of maximum flow in a planar network.
Abstract: We introduce a new class of problems concerned with the computation of maximum flows through two-dimensional polyhedral domains. Given a polyhedral space (e.g., a simple polygon with holes), we want to find the maximum “flow” from a source edge to a sink edge. Flow is defined to be a divergence-free vector field on the interior of the domain, and capacity constraints are specified by giving the maximum magnitude of the flow vector at any point. Strang proved that max flow equals min cut; we address the problem of constructing min cuts and max flows. We give polynomial-time algorithms for maximum flow from a source edge to a sink edge through a simple polygon with uniform capacity constraint (with or without holes), maximum flow through a simple polygon from many sources to many sinks, and maximum flow through weighted polygonal regions. Central to our methodology is the intimate connection between the max-flow problem and its dual, the min-cut problem. We show how the continuous Dijkstra paradigm of solving shortest paths problems corresponds to a continuous version of Berge's algorithm for computation of maximum flow in a planar network.
30 citations
22 citations
1 Jan 2000
TL;DR: In this paper, a short alternative proof of the basic result on the structure of the unit ball of the polyhedral space (Theorem 1) and a related Theorem 2 is presented.
Abstract: A Banach space X is called polyhedral if the unit ball of each one of its finite-dimensional (equivalently: two-dimensional [6]) subspaces is a polytope. Polyhedral spaces were studied by various authors; most of the structural results are due to V. Fonf. We refer the reader to the surveys [1], [2] for other definitions of polyhedrality, main properties and bibliography. In this paper we present a short alternative proof of the basic result on the structure of the unit ball of the polyhedral space (Theorem 1) and a related Theorem 2.
17 citations
1 May 2021
TL;DR: In this article, it was shown that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property, and that the phase-isometry of a map is a linear isometry.
Abstract: We say that a map is a phase-isometry if the equality
\[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \]holds for all is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.
10 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 2 |
| 2017 | 2 |
| 2016 | 3 |
| 2015 | 6 |
| 2014 | 3 |