Let $d(S)$, the diameter of a set $S$ be the maximum (supremum) distance between any two points in $S$. For any given $d_0$ real number, what is $\max\{\mathrm{area}(S)|S\hbox{ is a measurable set on the plane }\land d(S)=d_0\}$?
(Maybe a restriction more practical than measurable can be used.)
I suspect that the maximum is $\pi\cdot (d_0/2)^2$ (i.e. the circle has maximum area). I'm looking for a proof (or for a link to a proof).
The first result for https://www.google.com/search?q=isodiametric%20inequality, The two-dimensional isodiametric inequality contains a short proof of my conjecture.
For 3 dimensions, it cites: The isodiametric inequality states that, of all bodies of a given diameter, the sphere has the greatest volume. A proof can be found, e.g., in Lawrence C. Evans & Ronald F Gariepy: Measure theory and fine properties of functions.