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While trying to prove that the convex hull of a set $S \in R^{n}$ is the smallest convex set containing S, I encountered some issues. However, i found a very useful answer in Prove that the convex hull of X is the smallest convex set containing X which provided a full solution.

However, i could not understand the very last sentence in the link, which says:

"Finally, to show that any convex set D that contains X must contain C(X), simply use that D is a convex set."

I just cannot figure out why this is true nor how to write it down. Another user also asked the same thing to the user that wrote the answer but he did not respond.

Relevant definitions:

The convex hull of $X$, $C(X)$ is defined as $$C(X)=\left\{\sum_{i=1}^{n} a_{i} x_{i} \mid a_{1}, \ldots, a_{n} \geq 0, \sum_{i=1}^{n} a_{i}=1\right\}$$ Furthermore:

A set $S$ in $R^{n}$ is convex if given $x_{1}$ and $x_{2}$ in S, then $\lambda x_{1}+(1-\lambda) x_{2}$, must also belong to $S$ for each $\lambda \in[0,1]$.

Can someone help?

Thanks in advance, Lucas

Rodrigo de Azevedo
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Lucas
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    Since $D$ is convex and $D\supset X$, then if $\lambda 1,...,\lambda _n\in [0,1]$ are s.t. $\lambda _1+...+\lambda _n=1$, and $x_1,...,x_n\in X$, therefore $\sum{i=1}^n \lambda _ix_i \in D$. Therefore $D\supset C(X)$. – Surb Feb 14 '21 at 20:17
  • Is $\sum_{i=1}^{n} \lambda_{i} x_{i} \in D$ true because $x_{1}, \ldots, x_{n} \in X$ and, since $D \supset X$, $x_{1}, \ldots, x_{n} \in D$? – Lucas Feb 14 '21 at 21:42
  • Also, i cannot see where the convexity of the set $D$ was used. Would you mind clarifying that to me please? – Lucas Feb 14 '21 at 21:43
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    $D$ convex $\iff$ for all $\lambda 1,...,\lambda _n\in [0,1]$ s.t. $\sum{i=1}^n \lambda i=1$ and for all $d_1,...,d_n\in D$ we have $\sum{i=1}^n \lambda _id_i\in D.$ – Surb Feb 14 '21 at 21:59
  • hmmm interesting... But the definition used by the book I am following includes $(1-\lambda)$ in the summation, whereas yours does not seem to do so. Can you clarify that? – Lucas Feb 14 '21 at 22:09
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    Both are equivalents. Try to prove it :-) – Surb Feb 14 '21 at 22:09

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