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, then x can be written as the convex combination of at most d + 1 points in P. Namely, 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] 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.[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 non-negative, and .

Suppose k > d + 1 (otherwise, there is nothing to prove). 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 Perron–Frobenius theorem.[5][6]

Generalizations[edit]

Colorful Carathéodory theorem[edit]

Let A1, ..., Ad+1 be sets in Rd and a1A1, ..., ad+1Ad+1. If these sets shares a common point a in their convex hulls, then there is a set T = {a1, ..., ad+1} such that the convex hull of T contains the point a.[7]

By viewing the sets A1, ..., Ad+1 as different colors, the set T is made by points of all colors, hence the "colorful" in the theorem's name.[8]

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. arXiv:1112.5942. doi:10.1007/s00454-012-9439-z. ISSN 0179-5376.
  2. ^ Danzer, L.; Grünbaum, B.; Klee, V. (1963). "Helly's theorem and its relatives". Convexity. Proc. Symp. Pure Math. 7. American Mathematical Society. pp. 101–179. See in particular p.109
  3. ^ Carathéodory, C. (1907). "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). Convexity. Cambridge University Press. doi:10.1017/cbo9780511566172. ISBN 9780511566172. See pages 40–41.
  6. ^ Naszódi, Márton; Polyanskii, Alexandr (2019). "Perron and Frobenius meet Carathéodory". arXiv:1901.00540 [math.MG].
  7. ^ Bárány, Imre (1982-01-01). "A generalization of carathéodory's theorem". Discrete Mathematics. 40 (2–3): 141–152. doi:10.1016/0012-365X(82)90115-7. ISSN 0012-365X.
  8. ^ Montejano, Luis; Fabila, Ruy; Bracho, Javier; Bárány, Imre; Arocha, Jorge L. (2009-09-01). "Very Colorful Theorems". Discrete & Computational Geometry. 42 (2): 142–154. doi:10.1007/s00454-009-9180-4. ISSN 1432-0444.

Further reading[edit]

  • Eckhoff, J. (1993), "Helly, Radon, and Carathéodory type theorems", Handbook of Convex Geometry, A, B, Amsterdam: North-Holland, pp. 389–448.

External links[edit]