Dittert conjecture

The Dittert conjecture asserts that, among convex bodies of fixed volume, a simplex maximizes the expected volume of the simplex spanned by independently sampled uniform points.

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Random simplex: A simplex whose vertices are selected randomly from a specified region or distribution. Its volume, averaged over independent uniform samples, is the quantity compared by the conjecture.
Affine independence: A set of points is affinely independent when none lies in the affine span of the others. Sampled points span a full-dimensional simplex only when they are affinely independent.
Uwe Dittert: A German mathematician associated with a conjecture on expected volumes of random simplices in convex bodies. The conjecture takes its name from the mathematician who formulated it.
Monte Carlo method: A computational method that estimates quantities by repeated random sampling. Sampling vertices repeatedly gives numerical estimates of the expected simplex volume.
Expected value: The probability-weighted average of a random variable's possible values. The conjectured extremum concerns an average over all sampled vertex configurations.
Euclidean space: A real vector space equipped with the standard inner product and its induced geometry. The conjecture is formulated for convex bodies and simplices in finite-dimensional Euclidean space.
Convex geometry: The branch of geometry studying convex sets and their properties. The conjecture is an extremal problem in this field.
Isoperimetric inequality: An inequality identifying shapes that extremize a geometric quantity under a fixed-size constraint. The conjecture is a simplex-extremal problem under fixed volume, akin to isoperimetric questions.
Uniform distribution: A probability distribution assigning equal density to every point in a specified region. It specifies how each vertex is sampled from the convex body.
Dimension: The number of independent coordinates needed to describe a space or geometric object. The number of sampled vertices and the meaning of simplex depend on the ambient dimension.