Sum of Distances in Normed Spaces

Abstract

A geometric proof for the following theorem due to Martelli and Busenberg is given. Integral geometry is used to discuss special cases and related results. Minkowski spaces are simply finite dimensional normed linear spaces. Smoothness assumptions on the boundary of the unit disk E for a Minkowski plane will enable us to use Crofton's simplest formula from integral geometry to give a proof for three points. If the unit ball for a 3-dimensional Minkowski space is a zonoid, then we used integral geometry for the case of four points forming a simplex. A zonoid is a limit of sums of segments. Bolker discusses equivalent conditions for a convex subset of Rn to be a zonoid.

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Document Details

Document Type
Technical Report
Publication Date
Oct 09, 1990
Accession Number
ADA238066

Entities

People

  • Mostafa Ghandehari

Organizations

  • Naval Postgraduate School

Tags

DTIC Thesaurus Topics

  • California
  • Differential Equations
  • Equations
  • Geometric Forms
  • Geometry
  • Inequalities
  • Integrals
  • Lines (Geometry)
  • Mathematics
  • Minkowski Space
  • Reflection
  • Three Dimensional
  • Triangles
  • Two Dimensional

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Graph Algorithms and Convex Optimization.

Technology Areas

  • Space