J. J. Sylvester’s two convex sets theorem and G.-L. Lesage’s theory of gravity. (English) Zbl 1313.60005
Summary: Given two convex sets \(K_1\) and \(K_2\) in the plane, J. J. Sylvester [Acta Math. 14, 185–205 (1890; JFM 22.0233.02)] computed the measure \(m(K_1,K_2)\) of the family of straight lines which meet both \(K_1\) and \(K_2\). As their distance \(d=d(K_1,K_2)\) increases to infinity
\[
\displaystyle {m(K_1,K_2)=h(K_1)h(K_2)/d+O(1/d^2)}
\]
for some \(h(K_1)\geq 0\) and \(h(K_2)\geq 0\), suggesting Newton’s law of attraction in the plane. We discuss the analogy in the spirit of G.-L. Lesage.
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
