I have a question about a answer in this post: convex hull specifically from the answer given by @Per Manne.

I know that $$B=\left\{ \begin{pmatrix} x \\ y \end{pmatrix}\in \mathbb{R}^{2}: y>0\right\}$$ is convex set, I could easily verify using the definition of convexity (in fact geometrically it is very intuitive), but how can I prove that $B$ is convex hull of $$A=\left\{\begin{pmatrix} x \\ y \end{pmatrix} \in \mathbb{R}^{2}: y \geq \frac{1}{1+x^{2}}\right\}?$$

I think that intuitively it is true, since it is known that by extensivity $A \subseteq conv(A):=B$.

But, by definition how can I prove that fact?


Let $(x,y) \in B$. Let $n$ be a positive integer such that $ n >|x|$ and $1+n^{2} >\frac 1 y $. Let $t=\frac {n-x} {2n}$. Then $0<t<1$. Now $(x,y)=t(-n,y)+(1-t)(n,y)$. Note that $(-n,y)$ and $(n,y)$ both belong to $A$. Hence any element of $B$ is a convex combination of two points of $A$.


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