I have seen various assertions that one can derive the isoperimetric inequality in the plane from Crofton's formula in geometric probability. Unfortunately, I have not managed to figure out such a proof (or find a reference containing said proof). Can anyone point me in the right direction?
A proof of the isoperimetric inequality using Crofton's formula is contained in these notes by Treibergs: http://www.math.utah.edu/~treiberg/isoperim/isop.pdf
See also the proof in Integral Geometry and Geometric Probability by Santalo.