Geometry in Hilbert spaces / spheres in high dimensions

Question

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are interested in whether or not the unit ball $$B_1 := \{h \in H \mid 1 \geq \|h\|_H\}$$ is a subset of the closed symmetric convex hull $\overline{\operatorname{sco}}(\alpha_n \, e_n \mid n \in \mathbb{N})$ of the series $(\alpha_n \, e_n)_{n \in \mathbb{N}} \subset H$. It is clear, that such a series exists, since we can iteratively construct one, such that for every $n \in \mathbb{N}$ the $n$-dimensional sphere $B_{1+\frac{1}{n}} \cap \langle e_1,...,e_n \rangle$ is in the convex hull $\overline{\operatorname{sco}}(\alpha_1 \, e_1,...,\alpha_{n} \, e_n)$.

My question is if there are any known bounds on the rate at which such a series $(\alpha_n)_{n \in \mathbb{N}}$ must tend to infinity, i.e. if one can construct the series in a way to have it grow relatively slowly.

I would also be interested in any useful literature and maybe a lower bound on the rate, other than $\mathcal{O}(1)$.

Any help is be much appreciated.

Answer 1

Set $c_n:=\frac{1}{\alpha_n}$. Clearly, for any $N>0$ one must have $\DeclareMathOperator{\spa}{span}$ $\DeclareMathOperator{\conv}{conv}$ $\newcommand{\bsB}{\boldsymbol{B}}$ $$ \bsB_1\cap\spa\{e_1,\dotsc, e_N\} \subset\conv\{ \pm\alpha_1 e_1,\dotsc,\pm\alpha_N e_n\}. $$ This happens if and only if $$ f_N(\alpha):=\min\{ x_1^2+\cdots +x_N^2;\;\;c_1x_1+\cdots +c_Nx_N=1\}\geq 1. $$ (The hyperplane $c_1x_1+\cdots +c_Nx_N=1$ cuts the $k$-th axis at the point $\frac{1}{c_k}=\alpha_k$.) Using Lagrange multipliers we get $$ f_N(\alpha)=\frac{1}{c_1^2+\cdots +c_N^2}\geq 1. $$ Thus we need $$\sum_n\frac{1}{\alpha_n^2}=\sum_n c_n^2\leq 1. $$