Gruber P. Convex and Discrete Geometry

42 Convex Bodies
Convex Hulls
Given a set A in E d , its convex hull, conv A, is the intersection of all convex sets
in E d which contain A. Since the intersection of convex sets is always convex,
conv A is convex and it is the smallest convex set in E d with respect to set inclusion,
which contains A. For the study of convex hulls we need the following concept: Let
x1,...,xn ∈ E d . Then any point x of the form x = λ1x1 + ··· + λnxn, where
λ1,...,λn ≥ 0 and λ1 +···+λn = 1, is a convex combination of x1,...,xn.
Lemma 3.1. Let A ⊆ E d . Then conv A is the set of all convex combinations of points
of A.
Proof. First, we show that
(1) The set of all convex combinations of points of A is convex.
Let x = λ1x1 + ··· + λmxm and y = λm+1xm+1 + ··· + λnxn be two convex
combinations of points of A and 0 ≤ λ ≤ 1. Then
(1 − λ)x + λy = (1 − λ)λ1x1 +···+(1 − λ)λmxm + λλm+1xm+1 +···+λλnxn.
Since the coefficients of x1,...,xn all are non-negative and their sum is 1, the
point (1 − λ)x + λy is also a convex combination of points of A, concluding the
proof of (1).
Second, the following will be shown, compare the proof of Jensen’s inequality,
see Theorems 1.9 and 2.1.
(2) Let C ⊆ E d be convex. Then C contains all convex combinations of its
points.
It is sufficient to prove, by induction, that C contains all convex combinations of any
n of its points, n = 1, 2,... This is trivial for n = 1. Assume now that n > 1 and that
the statement holds for n − 1. We have to prove it for n. Letx = λ1x1 +···+λnxn
be a convex combination of x1,...,xn ∈ C. Ifλn = 0, then x ∈ C by the induction
hypothesis. If λn = 1, then trivially, x = xn ∈ C. Assume finally that 0