How to prove that lattice width is attained?

Question

Let $K \subseteq R^n$ be a convex body, that is, a closed bounded convex set. Given a vector $d \in R^n$ we define the width of $K$ along $d$ to be:
$w_d(K) = \underset{x \in K}{max} \space d^tx - \underset{x \in K}{min} \space d^tx$

The lattice width of $K$ is defined as the minimum width along any integral vector $d$, that is
$w(K) = \underset{d \in Z^n}{min} \space w_d(K)$

How can we prove that this minimum is always attained when $K$ is full dimensional (i.e. the width of the lattice is bounded)?

Answer 1

The answer is mostly in Proposition 2.3. of:

Draisma, J., McAllister, T. B., & Nill, B. (2012). Lattice-Width Directions and Minkowski's 3rd-Theorem. SIAM Journal on Discrete Mathematics, 26(3), 1104-1107.