Problems about the convex hull.

convex-analysisconvex-hulls

I'm stuck in two problems concerning about convex hull.

  1. Let $A,B,C \not= \emptyset$, compact sets in $\mathbb{R^n}$. Show that if $A+B=A+C$ then $\text{conv}(B)=\text{conv}(C)$
  2. Let $B\not= \emptyset$ in $\mathbb{R^n}$. Show that $n \text{conv}(B)+B=(n+1)\text{conv}(B).$ Also show that this is not true if we take some $m<n$.

I would appreciate some tip.

Best Answer

This is too long for a comment, so I'm posting it as a (partial) answer. In general, it is true that $\text{conv}(A+B)=\text{conv}(A)+\text{conv}(B).$ And this is enough to prove the first part:

$t(A+B)+(1-t)(A+B) = (tA+(1-t)A) + (tB+(1-t)B) = A+B$, so the sum $A+B$ is convex, and since it also contains $A+B$, it must contain the convex hull of $A+B.$

On the other hand, if $x\in \text{conv}(A+B),$ then it is a finite sum $\sum t_i(a_i+b_i)$ for some $a_i\in A;\ b_i\in B;\ t>0;\ \sum t_i=1.$

Then, $\sum t_i(a_i+b_i)=\sum t_ia_i+\sum t_ib_i$ so $x\in \text{conv}(A)+\text{conv}(B).$

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