# Problems about the convex hull.

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.

I would appreciate some tip.

• Have you tried anything? This is not hard to show by taking, say, $b_1,b_2 \in B, \theta \in [0,1], b_3 = \theta b_1 + (1-\theta) b_2 \in conv(B)$ and argue from there. You just need to make the argument that this holds for any $b_1,b_2,b_3,\theta$ Feb 12, 2019 at 18:25
• I don't think that 1. is true: If you take $A = \mathbb R^n$ and $B,C$ arbitrary, you get $A + B = A = A + C$ for free. But $\operatorname{conv}(B) = \operatorname{conv}(C)$ fails for arbitrary sets.
– gerw
Feb 13, 2019 at 6:41
• @gerw I missed one hypothesis, thanks. Feb 14, 2019 at 6:55

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 $$x\in \text{conv}(A)+\text{conv}(B).$$