Let $X_d$ denote the space of unimodular lattices, equipped with the topology where the convergence is defined by the convergence of basis ($x_n\to x$ in $X_d$ $\iff$ there exist basis of $x_n$, say $(e_n^i)_{i=1}^d$, that converge to some basis $(e^i)_{i=1}^d$ of $x\in X_d$).
Likewise, let $Y_d$ denote the space of unimodular grids, which by definition are the translates of unimodular lattices ($x+v$ where $x\in X_d, v\in Y_d$).
Let $\pi:Y_d \to X_d, x+v \to x$ be the natural projection.
For $y\in X_d$, let $$P(y):=\{|\Pi_i w_i|: w=(w_1,\dots,w_d)\in Y_d\},$$
namely the sets of all absolute values of products of components of vectors in $Y_d$
Now we define a function $\mu:X_d \to \mathbb R$ as follows:
$$\mu(x):=\sup_{y\in \pi^{-1}(x)} \inf P(y).$$
I wonder how to prove the following elementary assertion:
$(*)$ If $x_n \to x$ in $X_d$, then $\limsup \mu(x_n) \le \mu(x)$.
My question comes from the paper http://doi.org/10.4007/annals.2011.173.1.1 by Uri Shapira (Ann. of Math. 2011 Remark 4.4), where it was stated the the assertion above follows from the following elementary lemma:
Lemma 4.2 (inheritance) If $y, y_0 ∈ Y_d$ are such that $y_0 ∈ Ay$, then $\overline{P(y_0)} ⊂ \overline{P(y)}$.
Here $A$ denotes the subgroup of $SL(d,\mathbb R)$ consisting of all diagonal matrices of positive entries. But I don't see what role of $A$ plays in the proof of $(*)$ above. But $(*)$ is also hard to prove directly, too.