Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T00:04:48.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Measure generation by Euler functionals

Part of: Geometric probability and stochastic geometry Combinatorial probability 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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

References

[1] Alexander, R. (1978) Planes for which the lines are the shortest paths between points. Illinois J. Math. 22, 177–190.Google Scholar
[2] Ambartzumian, R. V. (1974) A note on pseudo-metrics on the plane. Z. Wahrscheinlichkeitsth. 29, 25–31.CrossRefGoogle Scholar
[3] Ambartzumian, R. V. (1982) Combinatorial Integral Geometry. Wiley, New York.Google Scholar
[4] Ambartzumian, R. V. (1990) Factorization Calculus and Geometric Probability. Cambridge University Press.Google Scholar
[5] Ambartzumian, R. V. (1992) Remarks on measure generation in the space of lines in 3 (in Russian). Izv. Akad. Nauk Armenii, Matematika 27(5), 1–21.Google Scholar
[6] Aramian, R. (1992) Recovering the rose of directions from flag densities in 3 (in Russian). Izv. Akad. Nauk Armenii, Matematika 27(5), 22–35.Google Scholar
[7] Oganian, V. K. and Abdallah, A. (1992) On generation of measures in the space of lines by Finsler metrics (in Russian). Izv. Akad. Nauk Armenii, Matematika 27(5), 69–80.Google Scholar
[8] Pogorelov, A. V. (1979) Hilbert's Fourth Problem. Winston, New York.Google Scholar
[9] Sukiasian, H. S. (1993) On measure generation by spatial flag functions (in Russian). Izv. Akad. Nauk Armenii, Matematika 28(2), 61–70.Google Scholar

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