Inequalities for the curvature of curves and surfaces

ABSTRACT

In this paper, we bound the difference between the total mean curvatures of two closed surfaces in R³ in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.

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INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS
      G.2.1 Combinatorics
          Subjects: Counting problems

General Terms:
Algorithms, Theory

Keywords:
Fréchet distance, approximation, bottleneck distance, curvature, integral geometry, persistence diagrams, stability