Gruber P. Convex and Discrete Geometry

Mechanical Interpretation of the Convex Hull
3 Convex Sets, Convex Bodies and Convex Hulls 43
This lemma says that the convex hull of a finite set {x1,...,xn} in E d consists of all
centres of gravity of (non-negative) masses λ1,...,λn at the points x1,...,xn.
In a hand-written appendix to his third proof of the fundamental theorem of
algebra, which was printed in Werke 3, p.112, Gauss [363] gave an alternative
description: the convex hull of x1,...,xn consists of all points x whichhavethe
following property: assume that x supports positive mass, then there are masses at
the points x1,...,xn, such that x is in equilibrium with respect to the gravitational
pull exerted by the masses at x1,...,xn.
Carathéodory’s theorem [190]
refines the above lemma:
Theorem 3.1. Let A ⊆ E d . Then conv A is the set of all convex combinations of
affinely independent points of A, i.e. the union of all simplices with vertices in A.
We reproduce a proof due to Radon [821], see also Alexandroff and Hopf [9], p. 607.
Proof. Let x ∈ conv A. By Lemma 3.1 we may represent x in the form x = λ1x1 +
···+λnxn, where x1,...,xn ∈ A, λ1,...,λn > 0, λ1 +···+λn = 1 and n is
minimal. We have to show that the points x1,...,xn are affinely independent.
Assume not. Then there are numbers µ1,...,µn, not all 0, such that
(3) µ1 +···+µn = 0,
(4) µ1x1 +···+µnxn = o.
By (3) at least one µk is positive. Choose k such that λk/µk is minimal among all
such k. Then
λi − λk
µk
�
λ1 − λk
µ1
µk
by (3). Hence (4) implies that
µi ≥ 0fori = 1,...,n, λk− λk
µk = 0,
�
µk
�
+···+ λn − λk
�
µn = 1
µk
x = λ1x1 +···+λnxn = λ1x1 +···+λnxn − λk
(µ1x1 +···+µnxn)
=
�
λ1 − λk
� �
µ1 x1 +···+ λk−1 −
µk
λk
µk
+···+
µk−1
µk
�
λn − λk
µn
µk
�
xk−1 +
�
xn
�
λk+1 − λk
�
µk+1 xk+1
µk
is a representation of x as a convex combination of at most n − 1 points of A. This
contradicts our choice of n and thus concludes the proof. ⊓⊔
Remark. Carathéodory’s theorem is a cornerstone of combinatorial convex geometry.
For more information on Carathéodory’s theorem and, more generally, on combinatorial
geometry, see Sect. 3.2 and the surveys and books cited there.