The Blaschke–Santaló bounds for the Steklov–Neumann functional

Let $\Omega\subset\mathbb{R}^2$ be a bounded, convex, open set, and let $F(\Omega)$ denote the functional introduced in the paper.

Blaschke–Santaló bounds. For every such $\Omega$,

$$1\leq F(\Omega)\leq 2.$$

These bounds would describe the conjectured range of the functional for convex planar domains; the paper presents numerical evidence for them, with collapsing rectangles asymptotically approaching the lower value and collapsing triangles asymptotically approaching the upper value.