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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)?

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  • $\begingroup$ What have you tried? Can you prove that $w_d(K)$ is continuous as a function of $d$? Can you restrict '$d \in \mathbb{Z}^n$' some bounded set? $\endgroup$
    – ProAmateur
    Feb 26, 2018 at 1:01
  • $\begingroup$ $d$ is given. The explanation can not be changed. $\endgroup$ Feb 26, 2018 at 1:07
  • $\begingroup$ On second thought, why isn't the minimum value simply attained at $d = 0$? $\endgroup$
    – ProAmateur
    Feb 26, 2018 at 1:24
  • $\begingroup$ Nope, I don't think you are on the right track. $\endgroup$ Feb 26, 2018 at 16:32

1 Answer 1

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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.

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