这个作业是完成有界连通集、非凸多边形等数学问题

MATH/ECON 530
Homework Set 2

1.让Xi是R2中的有界连通集
对于我= 1,2,。 。 。 ,米 而且,让所有
Xi不相交:
Xi∩Xj =∅,i 6 = j,
让X∪成为所有Xi的联合

X∪=
[米
i = 1


首先,让m = 2。
(a)(10分)表明每个x∈conv(X∪)可以写成凸
X∪的两个边界点的组合。
(b)(4分)提供一个反例以表明(a)中的结果不
持有若Xi之一
未连接。
(c)(2分)使用对(b)的答案表明(a)中的结果不成立
如果m> 2。
2. (6 points) Let S be any closed, nonconvex polygon. Show that conv S has a
greater area than S, but a smaller perimeter.
3.
(a) (6 points) Prove that the closure of a convex set S is convex.
(b) (4 points) Provide a counterexample to show that the converse is not true.
4. Let X be a set of N points in Rn, N > n. From Caratheodory’s Theorem we
have that every z ∈ conv X can be written as a convex combination of m vectors
in X, m ≤ n + 1. Call any such set of m vectors X(z).
(a) (2 points) Find conditions (if any) such that the same set W = X(z) in
Caratheodory’s Theorem for every z ∈ conv X.
(b) (6 points) Find conditions (if any) such that the convex-combination representation of z in Caratheodory’s Theorem is unique for every z ∈ conv X.
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