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Tagged with sumset convex-analysis
6
questions
1
vote
1
answer
55
views
Affine Combinations and Span
I was reading a bit of convex analysis and came across this problem.
Let $S$ be convex. Let $A$ be the set of finite affine combinations of points in $S$ (i.e. finite linear combinations whose weights ...
- 11
0
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1
answer
79
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Is the sum of a closed unit ball and a closed set itself closed?
I am reading here that in a Banach space, the sum of the closed unit ball with a closed bounded convex set might fail to be closed itself. It seems there is a counterexample if and only if the ...
- 10.4k
0
votes
2
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414
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On the definition of the Minkowski sum
Closed sets need not be mapped to closed sets by the Minkowski sum as the following example shows:
$$ S_1 := \{ (x, y) \mid x, y \in \Bbb R, x \geq 0, x y \geq 1 \},$$
$$ S_2 := {\Bbb R} \times \{0\} $...
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14
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3
answers
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The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$.
Define a set $C$ given by
$$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
Is $C$ a convex set?
- 151
3
votes
1
answer
682
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What is first edge position in the Minkowski sum of two convex polygons in the plane?
I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here:
Then I tried to apply this method of the Minkowski sum in the example ...
- 131
3
votes
3
answers
1k
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Show by example that the Minkowski sum of two sets $X+Y$ may be convex even if neither $X$ nor $Y$ are convex
There were two parts to this question. I proved that the Minkowski sum of two sets $X+Y$ is convex whenever $X$ and $Y$ are convex, but how do I prove this second part? "Show by example that the ...
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