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 […]

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 […]

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 […]

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, […]

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

Intereting Posts

Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it
Relationship between complex number and vectors
What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q} \simeq \mathbb{Q}$ as fields.
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
Prove that $6|2n^3+3n^2+n$
Lebesgue integral and a parametrized family of functions
Writing the identity permutation as a product of transpositions
Egoroff's theorem in Royden Fitzpatrick (comparison with lemma 10)
Sum of a Hyper-geometric series. (NBHM 2011)
Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?
How to calculate gradient of $x^TAx$
Prove that lebesgue integrable equal lebesgue measure
Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
A nontrivial everywhere continuous function with uncountably many roots?
$k/(x^n)$ module with finite free resolution is free