Articles of convex geometry

The origin is not in the convex hull $\Rightarrow$ the set lies in a hemisphere?

I am trying to understand the proof of the following claim: Let $f:A \subseteq \mathbb{S}^n \to \mathbb{S}^n$ be an $L$-Lipschitz* map (with $L <1$). Then $f(A)$ is contained in the interior of a hemisphere. *The distance on $\mathbb{S}^n$ can be either the intrinsic one or the extrinsic (Euclidean) one, it does not matter. In the […]

Every closed convex cone in $ \mathbb{R}^2 $ is polyhedral

I recently came across the following problem: Let $ K \subseteq \mathbb{R}^2 $ be a closed convex cone (meaning K is closed under non negative linear combinations) and I am asked to show it is polyhedral meaning it is the intersection of finitely many half spaces, or alternatively that it is finitely generated $ K […]

Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I’m not sure how to approach this problem. It is easily proven that such a triangle should have its vertices on the perimeter of the polygon, but I don’t […]

Can any two disjoint nonempty convex sets in a vector space be separated by a hyperplane?

Let $V$ be a normed vector space over $\mathbb{R}$, and let $A$ and $B$ be two disjoint nonempty convex subsets of $V$. A geometric form of Hahn-Banach Theorem states that $A$ and $B$ can be separated by a closed hyperplane (i.e. there is $f \in V^\ast$ and $\alpha \in \mathbb{R}$ such that $f(a) \le \alpha, […]

$n$ points are picked uniformly and independently on the unit circle. What is the probability the convex hull does not contain the origin?

This question already has an answer here: Centre in N-sided polygon on circle 1 answer