3 problems

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{R}^d$ satisfying … A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and…

Let $X_1,\ldots, X_n$ be iid random vectors in $\mathbb{Z}^d$. A choice of weights $w_i\in \\{-1,1\\}$ should exist such that, for all non-zero $a_i\in \mathbb{R}$ and all…

Let $X$ be a normalized uniformly distributed random vector. For mutually unbiased bases, let $m_1,m_2,\ldots,m_k$ denote the relevant maximum second-moment quantities, and let…