Geometry in Hilbert spaces / spheres in high dimensions

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.

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 $$\sum_n\frac{1}{\alpha_n^2}=\sum_n c_n^2\leq 1.$$