Suppose I had T points in the interval $[0,1]$. Call them $e_1, \dots, e_T$.

Question 1: What is a good nontrivial bound on the geometric mean of $$\{|e_i - e_j| : 1 \leq i < j \leq T \}, $$ as a function of $T$, independent of our choice of $e_i$?

Question 2: Suppose we can choose $i$ that minimizes the geometric mean of $$E_i = \{ |e_i-e_j| : 1 \leq j \leq T, j \neq i \}.$$What is a good nontrivial upper bound on the geometric mean of $E_i$ as a function of $T$?

Some context:

Suppose I were given an arithmetic circuit that computes $$f(z) = \sum_{i=1}^t c_tz^{e_t},$$ with bounds $D$ on the degree and $T$ on the number of terms, and we would like to test whether $f$ is zero. One way to test this for $f$ over an arbitrary field is to compute images $f \bmod (z^p-1)$ for a set comprised of the smallest primes $p$ whose product exceeds $$\prod_{j \neq 1} (e_1 - e_j),$$ though one could replace $e_1$ with any of the $e_i$'s. A naive upper bound on this product is $D^T$. An answer to either of the questions above could improve this bound.