Lower bound on all pairwise dot products on a set of vectors

Question

Say we have a set of vectors ${x_1,...,x_n}\in \mathbb{R}^d$. Are there any inequalities that give a lower-bound on the set of pairwise dot-products on this set, i.e, some $c$ such that $$\forall i,j \quad c\leq x_i^Tx_j$$

For example, one I found using the Cauchy-Schwarz inequality is $$c=-\max_i\Vert x_i \Vert^2$$ But I am looking for other inequalities if there are any.

Answer 1

The Cauchy-Schwartz lower bound is sharp because your set may contain two vectors $x_{i}=(a_{1},\dots,a_{d})$ and $x_{j}=(-a_{1},\dots,-a_{d})$.

Answer 2

Actually Cauchy-Schwarz gives you $-m_1m_2$ as lower bound where the multi-set $\{\!\{m_1,m_2\}\!\}$ is that of the two largest values among the norms of the the $x_i$. But if there is an $x_i$ with maximal norm for which $-x_i$ is also among the vectors, then $m_1=m_2$ is gives the same lower bound as Cauchy-Schwarz, and this bound is attained. So it is not clear what kind of expression you would be looking for to get a better bound.