Carathéodory's theorem (convex hull)

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An illustration of Carathéodory's theorem for a square in R2

In convex geometry, Carathéodory's theorem states that if a point x of Rd 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 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.[1]

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.[2]

The result is named for Constantin Carathéodory, who proved the theorem in 1907 for the case when P is compact.[3] In 1914 Ernst Steinitz expanded Carathéodory's theorem for any sets P in Rd.[4]

Example[edit]

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.

Proof of Carathéodory's theorem[edit]

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 strictly positive, and .

Suppose k > d + 1 (otherwise, there is nothing to prove). Then, the points 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.

An alternative proof uses Helly's theorem.[5]

See also[edit]

Notes[edit]

  1. ^ Bárány, Imre; Karasev, Roman (2012-07-20). "Notes About the Carathéodory Number". Discrete & Computational Geometry. 48 (3): 783–792. doi:10.1007/s00454-012-9439-z. ISSN 0179-5376. 
  2. ^ Danzer et. al. (1963), p.109
  3. ^ Carathéodory, C. "Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen". Mathematische Annalen (in German). 64 (1): 95–115. doi:10.1007/BF01449883. ISSN 0025-5831. MR 1511425. 
  4. ^ Steinitz, Ernst (1913), "Bedingt konvergente Reihen und konvexe Systeme", J. Reine Angew. Math., 143 (143): 128–175, doi:10.1515/crll.1913.143.128. 
  5. ^ Eggleston, H.G. (1958) pp.40-41

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