Euler characteristic and related measures for random geometric sets

Abstract

By an elementary calculation we obtain the exact mean values of Minkowksi functionals for a standard model of percolating sets. In particular, a recurrence theorem for the mean Euler characteristic recently put forward is shown to be incorrect. Related previous mathematical work is mentioned. We also conjecture bounds for the threshold density of continuum percolation, which are associated with the Euler characteristic.

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Mecke, K.R., Wagner, H. Euler characteristic and related measures for random geometric sets. J Stat Phys 64, 843–850 (1991). https://doi.org/10.1007/BF01048319

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Key words

  • Random sets
  • integral geometry
  • Minkowski functionals
  • Euler characteristic
  • continuum percolation