In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i\_1,\dots,i\_L\} \in \{1,\dots,n\}$, where $L$ is a fixed integer less than $n$. Can this be done using the Sudakov-Fernique inequality? 
$$
\mathbb{E}\_{h} \min \_{i\_{1}, \ldots, i\_{L}}\left[\sum\_{j\neq i\_1,\dots,i\_L}h\_j\right].
$$



Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality
Your question is equivalent to asking about the expectation of the max, and asking for an upper bound on the expectation of the max. You have a bunch of Gaussians $\{Y\_\alpha\}\_{\alpha\in I}$ indexed by
a set $I$ of cardinality $M$ which equals $n$-choose-$L$. The variance of each is $n-L$, and the correlation
is at least $n-2L$ (better bounds can be given if $L$ is not small compared to $n$, and in particular if $L\geq n/2$, but I'll leave that to you). Hence, $E(Y\_\alpha-Y\_\beta)^2\leq 2(n-L)-2(n-2L)=2L$ if $\alpha\neq \beta$. Hence, by Sudakov-Fernique, the
expected maximum is bounded above by the expected maximum of a bunch of Gaussians $\{Z\_\alpha\}\_{\alpha\in I}$
with variance $n-L$ and pairwise covariance $n-2L$. But you can realize the $Z\_\alpha$'s as $$Z\_\alpha=\sqrt{n-2L} G+ \sqrt{L} W\_\alpha$$
where $G$ is a standard normal and the $W\_\alpha$ are iid. So the expected max of the $Z\_\alpha$ is the 
expected max of the $\sqrt{L}W\_\alpha$, ie roughly $\sqrt{2L\log M}$ (for $M$ large and $L<n/2$).


