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We say a subset $A$ of $\mathbb{R}^N$ is balanced if \begin{equation} x \in A, \lambda \in [-1,1] \implies \lambda x \in A. \end{equation} Given a subset $A$ of $\mathbb{R}^N$, we write \begin{...

Let $X$ be random vector on the unit-sphere $S_{n-1}$ in $\mathbb R^n$. We don't assume that the distribution of $X$ is uniform on $S_{n-1}$ I'm interested in proving the existence of a (...