Integral Equation between the Distribution of Sizes of Corpuscles in a Solid and the Distribution in its sections by k-planes

MOLTER, URSULA M.

Homenatge al professor Lluis Santalo i. Sors

Servicio de Publicaciones = Universitat de Girona

Lugar: Girona; Año: 2002; p. 81 - 89

In this article, wee consider a convex $n$-dimensional body $Q$ that contains randomly distributed $h$-dimensionalcorpuscles similar to a fixed one $K$, and is intersected by linear manifolds $E_k$, with$h+k \geq n$. We find an integral equation which relates the number of corpuscles -per unit-volume of $K$ - with similitude ratio $\lambda$  to thenumber of sections - per unit volume of $E_k$ - with $h+k-n$-dimensional volume $\sigma$.In particular we consider the case of $h$-dimensional spherical corpusclesintersected by $k$-planes.For $h=n$, this extends the results of Kendall-Moran~\cite{KM63}, Santal\'o~\cite{San55} and Wicksel~\cite{Wic25} to the $n$-dimensional euclidean space $E_n$.