Carathéodory's theorem (convex hull)

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 x, 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.

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 :

${\displaystyle \mathbf {x} =\sum _{j=1}^{k}\lambda _{j}\mathbf {x} _{j}}$

where every x_j is in P, every λ_j is (w.l.o.g.) positive, and ${\displaystyle \sum _{j=1}^{k}\lambda _{j}=1}$.

Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the vectors x₂ − x₁, ..., x_k − x₁ are linearly dependent,

so there are real scalars μ₂, ..., μ_k, not all zero, such that

${\displaystyle \sum _{j=2}^{k}\mu _{j}(\mathbf {x} _{j}-\mathbf {x} _{1})=\mathbf {0} .}$

If μ₁ is defined as

${\displaystyle \mu _{1}:=-\sum _{j=2}^{k}\mu _{j}}$

then

${\displaystyle \sum _{j=1}^{k}\mu _{j}\mathbf {x} _{j}=\mathbf {0} }$

${\displaystyle \sum _{j=1}^{k}\mu _{j}=0}$

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

${\displaystyle \mathbf {x} =\sum _{j=1}^{k}\lambda _{j}\mathbf {x} _{j}-\alpha \sum _{j=1}^{k}\mu _{j}\mathbf {x} _{j}=\sum _{j=1}^{k}(\lambda _{j}-\alpha \mu _{j})\mathbf {x} _{j}}$

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

${\displaystyle \alpha$ :=\min _{1\leq j\leq k}\left\{{\tfrac {\lambda _{j}}{\mu _{j}}}:\mu _{j}>0\right\}={\tfrac {\lambda _{i}}{\mu _{i}}}.}

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

${\displaystyle \lambda _{j}-\alpha \mu _{j}\geq 0.}$

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

${\displaystyle \mathbf {x} =\sum _{j=1}^{k}(\lambda _{j}-\alpha \mu _{j})\mathbf {x} _{j}}$

where every ${\displaystyle \lambda _{j}-\alpha \mu _{j}}$ is nonnegative, their sum is one , and furthermore, ${\displaystyle \lambda _{i}-\alpha \mu _{i}=0}$. 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.

Alternative proofs uses Helly's theorem or the Perron–Frobenius theorem.[5][6]

• Shapley–Folkman lemma
• Helly's theorem
• Radon's theorem, and its generalization Tverberg's theorem
• Krein–Milman theorem
• Choquet theory

• Eckhoff, J. (1993), "Helly, Radon, and Carathéodory type theorems", Handbook of Convex Geometry, A, B, Amsterdam: North-Holland, pp. 389–448.
• Mustafa, Nabil; Meunier, Frédéric; Goaoc, Xavier; De Loera, Jesús (2019). "The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg". Bulletin of the American Mathematical Society. 56 (3): 415–511. arXiv:1706.05975. doi:10.1090/bull/1653. ISSN 0273-0979.

• Concise statement of theorem in terms of convex hulls (at PlanetMath)