On the Size of Approximately Convex Sets in Normed Spaces (Preprint)

Abstract

Let X be a normed space. A set A subset of X is approximately convex if d(ta + (1 - t)b,A) is less or equal to 1 for all a, b elements of A and t element of [0, 1]. We prove that every n-dimensional normed space contains approximately convex sets A with H(A, Co(A)) greater or equal to log2 n - 1 and diam(A) smaller or equal to C(square root of n)(ln n)squared where H denotes the Hausdorff distance. These estimates are reasonably sharp. For every D >0, we construct worst possible approximately convex sets in C(0, 1) such that H(A, Co(A)) = diam(A) = D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

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Document Details

Document Type
Technical Report
Publication Date
Aug 16, 1999
Accession Number
ADA640716

Entities

People

  • James W. Roberts
  • Ralph Howard
  • S. J. Dilworth

Organizations

  • University of South Carolina

Tags

Communities of Interest

  • Energy and Power Technologies

DTIC Thesaurus Topics

  • Abstracts
  • Banach Space
  • Convex Sets
  • Diameters
  • Equations
  • Geometry
  • Hilbert Space
  • Inequalities
  • Information Operations
  • Intervals
  • Mathematical Analysis
  • Mathematics
  • Notation
  • Probability
  • Random Variables
  • Step Functions
  • Theorems

Fields of Study

  • Mathematics

Readers

  • Analytical Mechanics
  • Graph Algorithms and Convex Optimization.

Technology Areas

  • Space