Buffon Needle Problem Over Convex Sets

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.