1. Convex Sets 5
Remark: See Section II.11 and, especially, Problem 3 of Section II.11.3.
Often, we consider convex sets in a more general setting.
(1.4) Convex sets in vector spaces. We recall that a set V with the operations
“+” (addition): V ×V −→ V and “·” (scalar multiplication): R×V −→ V is called
a (real) vector space provided the following eight axioms are satisfied:
(1) u + v = v + u for any two u, v V ;
(2) u + (v + w) = (u + v) + w for any three u, v, w V ;
(3) (αβ)v = α(βv) for any v V and any α, β R;
(4) 1v = v for any v V ;
(5) + β)v = αv + βv for any v V and any α, β R;
(6) α(v + u) = αv + αu for any α R and any u, v V ;
(7) there exists a zero vector 0 V such that v + 0 = v for each v V and
(8) for each v V there exists a vector v V such that v + (−v) = 0.
We often say “points” instead of “vectors”, especially when we have no par-
ticular reason to consider 0 (which we often denote just by 0) to be significantly
different from any other point (vector) in V .
A set A V is called convex, provided for all x, y A the interval
[x, y] = αx + (1 α)y : 0 α 1
is contained in A. Again, we agree that the empty set is convex. A convex combi-
nation of a finite set of points in V and a convex hull conv(A) of a set A V are
defined just as in the case of Euclidean space.
PROBLEM.
1◦.
Let V be the space of all continuous real-valued functions f : [0, 1] −→ R.
Prove that the sets
B = f V : |f(τ)| 1 for all τ [0, 1] and
K = f V : f(τ) 0 for all τ [0, 1]
are convex.
(1.5) Operations with convex sets. Let V be a vector space and let A, B V
be (convex) sets. The Minkowski sum A + B is a subset in V defined by
A + B = x + y : x A, y B .
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