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,

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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