All Questions

I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...

Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let $\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define $$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$ where ...

A subset $A$ of a metric space is called $\varepsilon$-separated if $$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$ (Notice that the inequality in my definition is strict.) What is the ...

Question: Does there exist a function $C:~(0,1)\to (0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space $X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point set $\{x_i\...