When S is a subset of an inner product space, let d(S) denote ${\sum\limits\_{s \in S} e^{- \langle s,s \rangle}}$


Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, and V is a subspace of $\mathbb{R^n}$.


Does this hold in general: ${\frac{d(L)} {d(L \cap V)}} \geq {\frac{d(M)}{d(M \cap V)}}$?


Why I am interested in this:
There are analogies (analogue of the Riemann Roch theorem for example) which imply it makes sense to think of metrized lattices as sheaves and the log of d as the dimension of the global sections of a sheaf(by "metrized lattice" I mean a lattice with an inner product).
So then given a map between metrized lattices, say f: A -> B, an analogous way to define "rank of induced map on global sections" would be:
log(d(A)) - log(d(ker(f))) (from the rank-nullity theorem). I will denote this quantity as rank(f).
So then if this rank function actually behaves as "rank of induced map on global sections", we would expect that $rank(g \circ f)$ is no greater than either rank(f) or rank(g). I am trying to show this for homomorphisms between metrized lattices that don't increase the norm at any point.
$rank(g \circ f) \leq rank(f)$ is trivial; $rank(g \circ f) \leq rank(g)$ is equivalent to this problem I have posed.


I am interested in this rank function because it seems to be possible to do derived functor cohomology without an actual left-exact functor but with just the
ranks of the maps that a left-exact functor would induce. Fully explaining myself here would result in an unreasonably big MO question.


$\textbf{Update:}$ The ideas mentioned in the previous paragraph, which are what originally motivated me to consider this question, are now fully explained [here](http://arxiv.org/abs/1509.05797). Oded and Noah's result is applied in section 7.



Inequality regarding sum of gaussian on lattices
Oded Regev and I just posted a [paper](http://arxiv.org/abs/1502.04796) to the arXiv that answers this question. See Corollary 4.4. In particular, we show that the OP's inequality does in fact hold and that it follows from a new inequality concerning the Gaussian mass of lattice cosets. 


Interestingly, we came to this question from a very different perspective; we were trying to improve upon the current fastest known algorithm for the shortest vector problem on lattices.


