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Carathéodory's theorem (convex hull)

See also Carathéodory's theorem for other meanings

In convex geometry Carathéodory's theorem states that if a point x of R^d lies in the convex hull of a set P, there is a subset P′ of P consisting of d+1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in a r-simplex with vertices in P, where

r leq d

. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Steinitz expanded Carathéodory's theorem for any sets P in R^d.

For example, consider a set P = {(0,0), (0,1), (1,0), (1,1)} which is a subset of R². The convex hull of this set is a square. Consider now a point x = (1/4, 1/4), which is in the convex hull of P. We can then construct a set {(0,0),(0,1),(1,0)} = P ′, the convex hull of which is a triangle and encloses p, and thus the theorem works for this instance, since |P′| = 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 :

mathbf{x}=sum_{j=1}^k lambda_j mathbf{x}_j

where every x_j is in P, every λ_j is positive, and $sum_{j=1}^klambda_j=1$ .

Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points x₂ − x₁, ..., x_k − x₁ are linearly dependent, so there are real scalars μ₂, ..., μ_k, not all zero, such that

sum_{j=2}^k mu_j (mathbf{x}_j-mathbf{x}_1)=mathbf{0}.

If μ₁ is defined as

mu_1:=-sum_{j=2}^k mu_j

then

sum_{j=1}^k mu_j mathbf{x}_j=mathbf{0}

sum_{j=1}^k mu_j=0

and not all of the μ_j are equal to zero. Therefore, at least one μ_j>0. Then,

mathbf{x} = sum_{j=1}^k lambda_j mathbf{x}_j-alphasum_{j=1}^k mu_j mathbf{x}_j = sum_{j=1}^k (lambda_j-alphamu_j) mathbf{x}_j

for any real α. In particular, the equality will hold if α is defined as

alpha:=min_{1leq j leq k} left{ tfrac{lambda_j}{mu_j}:mu_j>0right}=tfrac{lambda_i}{mu_i}.

Note that α>0, and for every j between 1 and k,

lambda_j-alphamu_j geq 0.

In particular, λ_i − αμ_i = 0 by definition of α. Therefore,

mathbf{x} = sum_{j=1}^k (lambda_j-alphamu_j) mathbf{x}_j

where every $lambda_j - alpha mu_j$ is nonnegative, their sum is one , and furthermore, $lambda_i-alphamu_i=0$ . 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.

Q.E.D.

References

C. Caratheodory, Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo, Vol. 32 (1911), 193-217.