# Meeting Details

Title: The Perimeter of a Convex Region Slow Pitch Seminar Robert Foote, Wabash College The perimeter (circumference) of a circle is $2\pi r = \pi d$, where $r$ and $d$ are the radius and diameter of the circle. Cauchy generalized this by showing that the circumference of a convex region in $\mathbb R^2$ is $2\pi \rho = \pi w$, where $\rho$ and $w$ are, in a certain sense, the average radius and width of the region. The proof involves only calculus and some simple geometric observations. The Spanish mathematician Santal\'o generalized this to spherical and hyperbolic geometry in two ways using the tools of integral geometry (a.k.a. geometric probability). I will give proofs that are in the spirit of, and nearly as simple as, the proof in $\mathbb R^2$. In addition, the proofs will be valid in neutral'' geometry, that is, Euclidean, spherical, and hyperbolic geometry all together. Some results generalize to an arbitrary surface.