I am trying to find a distribution over $n$ random vectors, say $x_1,\ldots, x_n$, on the $k$-dimensional unit sphere (where $n > k$) that minimizes $\max_{i\neq j} \mathrm{Var}(x_i^T x_j)$ subject to the constraint $\mathbb{E}[x_i^Tx_j]=0$.I tried some distributions and almost all them have variance $1/k$. For example, both the distribution in which each coordinate of each $x_i$ is independently and uniformly chosen from $\left\{-1/\sqrt{k}, 1/\sqrt{k}\right\}$ and the distribution in which each $x_i$ is an independent uniform vector on the $k$-dimensional unit sphere have variance $1/k$.Is $1/k$ the minimum variance among all distributions?
What is the minimum over all distributions of unit vectors of the variance of the dot product of the vectors?
reference request;randomized algorithms