Ambartzumian combinatorial formula and its applications

2023.9.1 19:30 N613

Consider a finite set of segments on a plane with endpoints in the general position. In [1] Sylvester posted a problem of finding the measure of the set of lines intersecting all of this segments. In [2] Ambartzumian obtained combinatorial formula that gives an answer to the posted problem. Ambartzumian also gave a combinatorial proof of the Pleijel identity ([3]) by integrating the aforementioned formula. We will discuss proof of Ambartzumian formula and show how it can be used to prove the Pleijel identity.

[1] Sylvester, J. J. On a funicular solution of Buffon’s “problem of the needle” in its most general form // Acta Mathematica, 14(1): 185–205, 1890.

[2] Ambartzumian, R.V. Combinatorial integral geometry: with applications to mathematical stereology. – John Wiley & Sons – 1982.

[3] Pleijel, A. Zwei kurze Beweise der isoperimetrischen Ungleichung. // Archiv der Mathematik, 7(4): 317–319, 1956.