sum of two points on the sphere. With additional assumptions on , upper bounds strictly lower than can be obtained. Dimensionless variant Recently, Adiprasito, Barany, Mustafa and Terpai proved a variant of Caratheodory's theorem that does not depend on the dimension of the space. Colorful Carathéodory theorem Let X1, ..., Xd+1 be sets in Rd and let x be a point contained in the intersection of the convex hulls of all these d+1 sets. Then there is a set T = {x1, ..., xd+1}, where x1 ∈ X1, ..., xd+1 ∈ Xd+1, such that the convex hull of T contains the point x. By viewing the sets X1, ..., Xd+1 as different colors, the set T is made by points of all colors, hence the "colorful" in the theorem's name. The set T is also called a rainbow simplex, since it is a d-dimensional simplex in which each corner has a different color. This theorem has a variant in which the convex hull is replaced by the conical hull. Let X1, ..., Xd be sets in Rd and let x be a point contained in the intersection of the conical hulls of all these d sets. Then there is a set T = {x1, ..., xd}, where x1 ∈ X1, ..., xd ∈ Xd, such that the conical hull of T contains the point x. Mustafa and Ray extended this colorful theorem from points to convex bodies. The computational problem of finding the colorful set lies in the intersection of the complexity classes PPAD and PLS. See also Shapley–Folkman lemma Helly's theorem Kirchberger's theorem Radon's theorem, and its generalization Tverberg's theorem Krein–Milman theorem Choquet theory Notes Further reading External links Concise statement of theorem in terms of convex hulls (at PlanetMath) Articles containing proofs Convex hulls Geometric transversal theory Theorems in convex geometry Theorems in discrete geometry