# Is the number of vertices of a convex $d-$dimensional lattice polytop without interior lattice points bounded?

The lattice polytop $[0,n_1]\times[0,n_2]\times\dots\times[0,n_{d-1}]\times[0,1]$ contains $(n_1+1)(n_2+1)\cdots(n_{d-1}+1)2$ integral points on the boundary and no integral points in its interior. Its number of vertices, $2^d$, is however bounded by a function depending only on its dimension $d$. Does there exist a sequence of convex $d-$dimensional lattice-polytops without interior lattice points and more and more vertices?

Remarks: (1) The answer is no in dimension $2$.

(2) This question is motivated by a result of Lagarias-Ziegler who showed that the volume (and thus the number of vertices) of a convex $d-$dimensional lattice polytop is bounded if it contains exactly $k>0$ interior lattice points. If no sequence as above exist, then the condition on the existence of $k$ interior lattice points can perhaps be modified into a condition on the number of integral vertices (which has to be sufficiently large) and integral boundary points.

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Why not take a polygonal prism? Depending on the polygon, you can get twice the number of vertices of the (d-1)- dimensional polygon. Or am I forgetting something? Gerhard "Ask Me About System Design" Paseman, 2012.02.09 – Gerhard Paseman Feb 9 '12 at 17:59
The more I look at the question, the more one thing becomes clear: at least one of us has not had enough coffee this morning. Gerhard "It's Quite Possible Its Me" Paseman, 2012.02.09 – Gerhard Paseman Feb 9 '12 at 18:08
Interestingly, it is known that there are only a finite number of maximal hollow lattice polytopes in a fixed dimension, where "maximal hollow" means "not properly contained in a larger hollow lattice polytope" (Averkov, Wagner, Weismantel; Nill & Ziegler). – Joseph O'Rourke Feb 9 '12 at 19:07
I think that ask me about system design provided an answer to the question: in dimensions ≥3, the number of vertices is not bounded. – André Henriques Feb 9 '12 at 19:54
Joseph, if it is not too much trouble, would you leave a comment enlightening me (and future readers) briefly on hollow polytopes? A rough idea or Wikipedia link would suffice. Gerhard "Enquiring Minds Want To Know" Paseman, 2012.02.09 – Gerhard Paseman Feb 9 '12 at 21:09