Abstract
I. Mean distance between two randomly selected points chosen uniformly from the interior of a given convex polyhedron $K$ was known in the exact form only for $K$ being a cube (the Robbins constant). However, a modification of the Crofton Reduction Technique always turns the problem into finite series of solvable double integrals. This way, we managed to derive the exact mean distance for all other regular polyhedra (tetrahedron, octahedron, dodecahedron, icosahedron). As the procedure can be done for any polyhedron, the mean distance is always expressible in the exact form.
II. The mean tetrahedron volume problem asks to find the expected volume of a convex hull of four points (which form a tetrahedron almost surely) selected uniformly and independently from the interior of $K$. In 90's (Buchta and Reitzner) and 00's (Zinani), Efron's formula was used to deduce the mean volume in the case of $K$ being a tetrahedron and a cube, respectively. Shortly after Christmass 2020, using the same Efron's formula with more optimal parametrisation and heavily relying on computer algebra system Mathematica, we extended the result for $K$ being a regular octahedron. The exact value turned out to be
$$\frac{19297\pi^2}{3843840}-\frac{6619}{184320} = 0.0136374112765241754602123153299677982932384778749528778646436210...$$
In subsequent months of 2021, we also found the exact mean tetrahedron volume in six other polyhedra: triangular prism, square pyramid, rhombic dodecahedron, cuboctahedron, triakis tetrahedron, truncated octahedron. In our presentation, we shortly outline the general method how we obtained these results and also briefly discuss how we could proceed in higher dimensions as the Efron's formula possesses a simple generalisation there.