8 I. Convex Sets at Large
m = 1, then u = u1 and u A since S A. Suppose that m 1. Then αm 1
and we may write
u = (1 αm)w + αmum, where w =
α1
1 αm
u1 + . . . +
αm−1
1 αm
um−1.
Now, w is a convex combination of u1,... , um−1 because
m−1
i=1
αi
1 αm
=
1
1 αm
m−1
i=1
αi =
1 αm
1 αm
= 1.
Therefore, by the induction hypothesis, we have w A. Since A is convex, [w, um]
A, so u A.
PROBLEMS.
1◦. Prove that conv
(
conv(S)
)
= conv(S) for any S V .
2◦. Prove that if A B, then conv(A) conv(B).
3◦.
Prove that conv(A) conv(B) conv(A B).
4. Let S V be a set and let u, v V points such u / conv(S) and
v / conv(S). Prove that if u conv
(
S {v}
)be
and v conv
(that
S {u}
)
, then u = v.
5 (Gauss-Lucas Theorem). Let f(z) be a non-constant polynomial in one com-
plex variable z and let z1,... , zm be the roots of f (that is, the set of all solutions
to the equation f(z) = 0). Let us interpret a complex number z = x + iy as a
point (x, y)
R2.
Prove that each root of the derivative f (z) lies in the convex
hull conv(z1, . . . , zm).
Hint: Without loss of generality we may suppose that f(z) = (z z1) · · · (z
zm). If w is a root of f (z), then
∑m
i=1 j=i
(w zj) = 0, and, therefore,
∑m
i=1 j=i
(w zj) = 0, where z is the complex conjugate of z. Multiply both
sides of the last identity by (w z1) · · · (w zn) and express w as a convex combi-
nation of z1,... , zm.
Next, we introduce two important classes of convex sets.
(2.2) Definitions. The convex hull of a finite set of points in
Rd
is called a
polytope.
Let c1,... , cm be vectors from
Rd
and let β1,... , βm be numbers. The set
P = x
Rd
: ci,x βi for i = 1, . . . , m
is called a polyhedron (see Problem 2 of Section 1.3).
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