Would $\Sigma_i^P \neq \Pi_i^P$ imply that polynomial hierachy cannot collapse to the $i$-th level?

If $ \Sigma_i^P = \Pi_i^P$ , then it follows that the polynomial hierarchy collapses to the $ i$ -th level.

What about the case $ \Sigma_i^P \neq \Pi_i^P$ ? For example, consider the case of $ NP \neq coNP$ . As far as I understand, this would imply the polynomial hierarchy cannot collapse to the first level, since if $ PH =NP$ , then in particular, $ coNP \subseteq NP$ , which means $ NP = coNP$ . Can we expand this idea to proof the general case: $ \Sigma_i^P \neq \Pi_i^P$ implies $ PH$ cannot collapse to $ i$ -th level?

Lowering $i$th shortest vector of a lattice

LLL guarantees that we can find a basis $ v_1,\dots,v_n$ of a lattice in $ \mathbb R^n$ with

$ $ \|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+1)}$ $ where $ \gamma_i$ is a function only of $ i$ and $ n$ .

  1. Are there lattices where this cannot be improved to $ \|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+2)}$ ?

  2. In general are there algorithms (possibly in exponential time) which can guarantee $ \|v_i\|\leq \gamma_{i,n} det(\Lambda)^{1/(n-i+2)}$ ?