Euclidean norms do not need full rank lattices

Question

There is a sentence in Micciancio's Lattice lectures that says when we bound the minimum distance of a lattice using the convex body theorem for norms other than the Euclidean norm we need to assume that they are full rank lattices. The reason stated for the full dimensionality condition not been necessary is that in Euclideadean norms one can embed any lattice in $\mathbb{R}^d$ of rank $n$ into $\mathbb{R}^n$ using orthogonal projections. My question is : are such orthogonal projections not possible in other norms?

Answer 1

Orthogonal transformations are closely related to the $\ell_2$ inner product. In fact, one can define the group of orthogonal matrices as the matrices $A$ such that: $$\langle A\vec a, A\vec b\rangle = \langle \vec a, \vec b\rangle$$ Note that this is equivalent to the "standard" definition $A^t A = I$.

Anyway, it is known that of all $\ell_p$ spaces, $\ell_2$ is the only normed space where the norm comes from an inner product, so while one can use the orthogonal (with respect to the $\ell_2$ inner product) transformations in whatever norm you want (as they are just matrices), there is no reason to think that their "nice" properties, such as $$\lVert x\rVert_2^2 = \lVert \pi_F(x)\rVert_2^2 + \lVert\pi_{F^\perp}(x)\rVert_2^2,$$ (where $\pi_F$ is the orthogonal projection onto the subspace $F$) will carry over to other norms.

This nice property seems key to me to the proof of what Micciancio is referring to. Namely for a lattice $\Lambda\subseteq \mathbb{R}^n$, if $E = \mathsf{span}_{\mathbb{R}}(\Lambda)$, then when working in the $\ell_2$ norm one can instead examine $\Lambda = \pi_E(\Lambda)\subseteq \pi_E(\mathbb{R}^n)\cong \mathbb{R}^{\mathsf{rk}(\Lambda)}$ instead. Note that this equation is always valid as an inclusion of sets, but we additionally care about metric details here.

I think geometrically what specifically goes wrong is as follows, but have not checked details. The Voronoi cell $V$ of a non-full rank lattice should not be a compact set (If $E = \mathsf{span}_{\mathbb{R}}(\Lambda)$, the cell should not be bounded on the set $E^\perp$). For the $\ell_2$ norm, the voronoi cell should still have relatively simple behavior on $E^\perp$ though --- intuitively it should be something like $V \times E^\perp$, as for arbitrary $x$ in space one can write $\lVert x\rVert_2^2 = \lVert \pi_E(x)\rVert_2^2 + \lVert \pi_{E^\perp}(x)\rVert_2^2$, and the closest lattice point to $x$ is entirely controlled by this first term. As this equation does not hold for more general norms, there is no reason for the Voronoi cells to exhibit this simple behavior, hence the restriction to being full-rank in the general case.