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Measure generation by Euler functionals

Part of: Combinatorial probability Geometric probability and stochastic geometry Miscellaneous topics in measure theory

Published online by Cambridge University Press:  01 July 2016

R. V. Ambartzumian
R. V. Ambartzumian*
Affiliation:
Armenian Academy of Sciences with an appendix by V. K. Oganian
*
* Postal address: Armenian Academy of Sciences, Erevan, Armenia.
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Abstract

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.

Keywords

SPHERICAL EXCESS FORMULANON-STANDARD MEASURE GENERATION

MSC classification

Primary: 28E05: Nonstandard measure theory
Secondary: 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 27 , Issue 3 , September 1995 , pp. 606 - 626
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research made possible in part by Grant no. RYBOOO from the International Science Foundation.

References

References

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Reference added in proof

[10] Ambartzumian, R. V., (Ed.) (1994) Analytical Implications of Combinatorial Integral Geometry. Izv. Akad. Nauk Armenii, Matematika [in Russian]. English translation: J. Contemp. Math. Anal (Armenian Academy of Sciences) 29(4), 1994.Google Scholar