I haven't seen a name for it, though something similar to it has been studied before. Let $\mathsf{KZ}(L)$ be the Korkin-Zolotarev basis of a lattice, which is a particular basis of the lattice with good geometric properties (much "higher quality" than basis output by something like LLL, but harder to compute). You can find a precise definition in the link below.
Lagarias, H. Lenstra, and Schnorr proved (Theorem 2.3) that
$$ \delta(\mathsf{KZ}(L))^2 \leq \gamma_n^n\prod_{i = 1}^n\frac{i+3}{4}, $$ where $\gamma_n = \sup_{L\in\mathcal{L}_n}\frac{\lambda_1(L)^2}{\sqrt[n]{\det L}^2}$ is Hermite's constant. This immediately implies the bound
$$ \delta_n \leq \sqrt{\gamma_n^n\prod_{i=1}^n \frac{i+3}{4}} $$ for your constant of interest.