Categories: Convex geometry | Discrete geometry | Theorems

Carathéodory's theorem (convex hull)

In mathematics Carathéodory's theorem on convex sets 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 no more than d+1 points such that x lies in the convex hull of P′. In other words, x lies in a d-simplex with vertices in P. The result is named for Constantin Carathéodory.

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 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 nonnegative, and $\sum_{j=1}^k\lambda_j=1$ .

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,

$\mathbf{x} = \sum_{j=1}^k \lambda_j \mathbf{x}_j-\alpha\sum_{j=1}^k \mu_j \mathbf{x}_j = \sum_{j=1}^k (\lambda_j-\alpha\mu_j) \mathbf{x}_j$

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

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

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

$\lambda_j-\alpha\mu_j \geq 0.$

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

$\mathbf{x} = \sum_{j=1}^k (\lambda_j-\alpha\mu_j) \mathbf{x}_j$

where every λ_j-αμ_j is nonnegative, their sum is one , and furthermore, $λ i - αμ i = 0$ . 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.

External links

Concise statement of theorem in terms of convex hulls (at PlanetMath)
Abstract for talk on theorem; abstract includes explanation of theorem in terms of convex hulls

Categories: Convex geometry | Discrete geometry | Theorems