When is the regular simplex the best shape? An Evidence Press briefing. Pick two random points from a set. If you may choose how they are distributed, how large can their average distance be? More generally, you can average a power of the distance, such as its square, and take the root at the end. Mathematicians call these quantities Riesz capacities with negative exponents. Comparing two such capacities, with two different powers, measures the shape of a set rather than its size. In 2024, Clark and Laugesen asked which shapes make that comparison as lopsided as possible, and they made two conjectures. This paper proves both. One way round, two points are always the extreme case, on any space with a distance. A short proof of this was already posted publicly a week earlier, and the paper says so. The other way round, for powers of two or more, the extreme shape in any dimension is the regular simplex: an equilateral triangle in the plane, a regular tetrahedron in space. Much of this second result also follows from earlier work by Davies, Lim and McCann, and the paper pins down exactly where. The genuinely new finding is that two is an exact threshold. For any power below two, you can beat the simplex by splitting one of its corners into two nearby points. So the simplex is the best shape if and only if the smaller power is at least two. The paper also measures how close a nearly extreme set must be to two points or to a simplex, with the best possible rates. The proofs are written out in full, and scripts check them on many examples, but the checks are not proofs. Three older sources could not be read. This is an anonymous, AI-assisted, unrefereed candidate, not peer reviewed or independently reproduced. Published on the third of November twenty twenty-six, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.