Carathéodory's theorem is a theorem in convex geometry. It states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset ′ of Pconsisting ofd + 1 or fewer points such that x lies in the convex hull of ′. Equivalently, x lies in an r-simplex with vertices in P, where. The smallest r that makes the last statement valid for each x in the convex hull of P is defined as the Carathéodory's number of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained. Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. The result is named for Constantin Carathéodory, who proved the theorem in 1907 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets P in Rd.
Example
Consider a set P = which is a subset of R2. The convex hull of this set is a square. Consider now a point x =, which is in the convex hull of P. We can then construct a set = ′, the convex hull of which is a triangle and encloses x, and thus the theorem works for this instance, since |′| = 3. It may help to visualise Carathéodory's theorem in 2 dimensions, as saying that we can construct a triangle consisting of points from P that encloses any point in P.
Proof
Let x be a point in the convex hull of P. Then, x is a convex combination of a finite number of points in P : where every xj is in P, every λj is positive, and. Suppose k > d + 1. Then, the vectors x2 − x1,..., xk − x1 are linearly dependent, so there are real scalars μ2,..., μk, not all zero, such that If μ1 is defined as then and not all of the μj are equal to zero. Therefore, at least one μj > 0. Then, for any real α. In particular, the equality will hold if α is defined as Note that α > 0, and for every j between 1 and k, In particular, λi − αμi = 0 by definition of α. Therefore, where every is nonnegative, their sum is one, and furthermore,. In other words, x is represented as a convex combination of at most k-1 points of P. This process can be repeated until x is represented as a convex combination of at most d + 1 points in P. Alternative proofs uses Helly's theorem or the Perron–Frobenius theorem.
If a point x of Rd lies in the conical hull of a set P, then x can be written as the conical combination of at most d points in P. Namely, there is a subset ′ of P consisting of d or fewer points, such that x lies in the conical hull of ′. The proof is similar to the original theorem; the difference is that, in a d-dimensional space, the maximum size of a linearly-independent set is d, while the maximum size of an affinely-independent set is d+1.
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.
Let X1,..., Xd+1 be sets in Rd and letx be a point contained in the intersection of the convex hulls of all these d+1 sets. Then there is a set T =, 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 rainbowsimplex, 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 =, where x1 ∈ X1,..., xd+1 ∈ Xd+1, such that the conical hull of T contains the point x. Mustafa and Ray extended this colorful theorem from points to convex bodies.
