Is difference of convex hulls a convex hull of differences?

Asked Sep 08 '15 at 15:33

Active Sep 08 '15 at 15:33

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Let $K \subset \mathbb{R}^n$ be an n-dimensional simplex. Let $e_i$ denote the $i^{th}$ standard unit vector. Define $K-K$ as follows: $$ K-K=\{x-y: x,y \in K\} $$ We know that $K=$ convex hull $\{e_1, e_2,\ldots, e_n , (0,0,0,...,0)\}$.

I am wondering if $K-K=$ convex hull of $ \{e_1,e_2,\ldots, e_n, (0,0,...,0) \} - \{e_1,e_2,\ldots, e_n, (0,0,...,0) \}$

This seems to be true for n=2 case.

asked Sep 08 '15 at 15:33

pikachuchameleon

2,889

As I understood, you ask whether $K-K$ is also a convex set ? – Svetoslav Sep 08 '15 at 15:45
Aaha, I think I understood your question – Svetoslav Sep 08 '15 at 15:47
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Apply the identity $\text{conv}(A+B)=\text{conv}(A)+\text{conv}(B)$. – A.Γ. Sep 08 '15 at 15:53
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For a proof of the identity .@A.G. mentioned, cf: http://math.stackexchange.com/questions/587162/prove-convex-hull-of-minkowski-sum – user251257 Sep 08 '15 at 16:23
@user251257: Thanks a lot. I forgot the simple identity – pikachuchameleon Sep 14 '15 at 03:24
@A.G: Thanks. I failed to figure out this simple observation. – pikachuchameleon Sep 14 '15 at 03:24

Is difference of convex hulls a convex hull of differences?

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