All Questions

I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as: Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...

Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ ...

Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...

The conjecture that I proposed in Maximal set on hypersphere that does not contain pairs of orthogonal vectors is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun. See for ...

Is the following fact true? Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...