12.6 Minkowski sum[2CP]
Let be in the following \(X\) be a vector space normed with norm \(\| ⋅\| \).
[11R] Let \(X\) be a vector space and \(A,B⊆ X\). We define the Minkowski sum \(A ⊕ B⊆ X \) as
In the following, given \(A⊆ X,z∈ X\) we will indicate with \(A+z=\{ b+z:b∈ B\} \) the translation of \(A\) in the direction \(z\).
- E345
[11S]Prerequisites:345.Show that the sum is associative and commutative; and that the sum has a single neutral element, which is the set \(\{ 0 \} \) consisting of the origin alone.
- E345
[11T] Prerequisites:345.If \(A\) is open, show that \(A⊕B\) is open. Hidden solution: [UNACCESSIBLE UUID ’11V’]
- E345
[11W]Prerequisites:345.If \(A,B\) are compact, show that \(A⊕B\) is compact. Hidden solution: [UNACCESSIBLE UUID ’11X’]
- E345
[11Y]Prerequisites:345.If \(A\) is a closed set and \(B\) is a compact set, show that \(A⊕B\) is closed. Hidden solution: [UNACCESSIBLE UUID ’11Z’]
- E345
[120]Prerequisites:345.Show an example where \(A,B\) are closed but \(A⊕B\) is not closed. Hidden solution: [UNACCESSIBLE UUID ’121’]
- E345
[122] Prerequisites:345.If \(A,B\) are convex show that \(A⊕B\) is convex. Hidden solution: [UNACCESSIBLE UUID ’123’]