这个Homework是完成集合相关的数学问题
MATH/ECON 530 Application of Mathematics to Economics
Homework Set 1
1.对于以下每个集合S∈R2
，请执行以下每个操作，说明
必要的子情况：
在不同情况下对应于α的不同区域的草图S
代数确定α的条件（如果有）以关闭S，
代数确定α的条件（如果有），其中S是凸的，
和
在S不为凸的情况下，绘制凸组合的线段
不包含在S中。
闭合证明不需要像凸度证明那么严格。
（a）（5分）S = {x≥0，y≥0，xy>α，α∈R}
（b）（9分）S = {y≥0，x2 + y
2 <1}
（c）（5分）S = {y≥1 +α| x |，α∈R}
2.（13分）考虑以下区域X：
X = {（x，y）| x≥0，0≤y≤f（x）}，f（x）∈C2
[0，b]，f（0）> 0，
其中b是f的最小正零。(Here C
2
[0, b] is the set of functions
with continuous second derivatives on [0, b].) Determine conditions on f(x) such
that X is convex.
3. (4 points) Let H = {x ∈ Rn|u
T x = c} for some given u ∈ Rn, c ∈ R. (Such a
set is called a hyperplane in Rn.) Show that H has no interior points.
4. (4 points) Either prove the following statement, or provide a general example
which disproves it. Again, just a diagram is not enough.
Let S1 and S2 be convex sets that intersect at infinitely many points with int
S1 6= φ, int S2 6= φ. Then int(S1 ∩ S2) 6= φ.
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