I want to prove the following statement: Given $A\subset \mathbb{R}^n$ let $C(A)$ be its convex hull. Prove that $\text{diam }(A)=\text{diam }(C(A))$. I can suppose that $A$ is a bounded closed set and I know that if $x,y\in A$ are such that $d(x,y)=\text{diam }(A)$ then $x,y\in \partial A$. I tried proving that if $z,w\in \partial C(A)$ […]

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 […]

The Calippo™ popsicle has a specific shape, that I would describe as a circle of radius $r$ and a line segment $l$, typically of length $2r$, that’s at a distance $h$ from the circle, parallel to the plane the circle is on, with its midpoint on a perpendicular line that goes through the centre of […]

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$, and I would like to check that some other point $y$ lies in their convex hull. How can I do this in some efficient way? I think that there was an algorithm based on checking the signs of pairwise inner products $x_i\cdot y$, however I was not able to […]

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

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm not greater than one? It is easy to show that a convex combination of orthogonal matrices has norm (I mean the norm as operators) not larger than $1$. The […]

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