Buffon Needle Problem Over Convex Sets

TL;DR

This paper investigates the probability that a randomly oriented needle of length l, originating within a convex set, remains entirely inside, and shows that the disk maximizes this probability among sets with equal perimeter.

Contribution

It introduces a novel variant of the Buffon Needle problem for convex sets and proves an isoperimetric inequality related to this probability.

Findings

The disk maximizes the probability among convex sets with the same perimeter.

An isoperimetric inequality is established using convex geometry techniques.

The probability depends on the perimeter and shape of the convex set.

Abstract

We solve a variant of the classical Buffon Needle problem. More specifically, we inspect the probability that a randomly oriented needle of length $l$ originating in a bounded convex set $X\subset\mathbb{R}^2$ lies entirely within $X$. Using techniques from convex geometry, we prove an isoperimetric type inequality, showing that among sets $X$ with equal perimeter, the disk maximizes this probability.