I am trying to properly write a proof of Minkowski's theorem in a self-contained way and understandable by (good) undergraduates.
Theorem (Minkowski)
Let $L$ be a lattice of $\mathbb{R}^n$ and $C$ a convex body, symmetric relatively to the origin and with $\mathrm{vol}{C} > 2^n \det(L)$. Then there is a nontrivial point of $L$ lying in $C$.
I would like to illustrate the use of Poisson formula and of analytic tools to approximate counting functions, for instance as a first contact with analytic number theory or trace formulas. Here are the steps :
1. Poisson formula
For $f$ continuous, with integrable Fourier transform, and both of moderate growth, $$\sum_{x \in L} f(x) = \mathrm{covol}(L)^{-1} \sum_{\xi \in \widehat{L}} \widehat{f}(\xi)$$
This follows from elementary Fourier analysis and is fine.
2. Procedure
We want to approximate the counting function of $C \cap L$ as a spectral side in the Poisson formula above. For this, we require a function $f$ such that
- $f \leqslant \mathbf{1}_C$
- $\widehat{f} \geqslant 0$
- $f$ admissible for Poisson formula
Provided such a function, the result then essentially follows from the computations $$|C \cap L| = \sum_{x \in L} \mathbf{1}_C(x) \geqslant \sum_{x \in L} f(x) = \mathrm{covol}(L)^{-1} \sum_{\xi \in \widehat{L}} \widehat{f}(\xi) \geqslant \mathrm{covol}(L)^{-1} \widehat{f}(0)$$
3. Construction of $\mathbf{f}$
Now it remains to be able to provide such an $f$. Naively the characteristic function of $C$ would work formally, but does not satisfy the continuity property.
In order to get positivity and regularity, $\mathbf{1}_{C/2}\star \mathbf{1}_{C/2}$ is a good try but does not respect the growth conditions in Poisson (at least it seems hard to prove it and it is not true in general, however I am ready to suppose any simplifying assumption on $C$ if it could help).
So here is the question: is there any simple construction of such a function $f$, or modification (e.g. mollifying) of the above ones?