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
习
。
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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