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Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to contain at least one point of the lattice (possibly on the boundary)?

The things I found on the internet all concern the case, in which it is already known, that at least one point of the lattice is contained in the region. Any suggestion would be welcome.

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  • $\begingroup$ I think this is a linear integer program that can be hard in general. $\endgroup$ May 29, 2016 at 23:15
  • $\begingroup$ What does it mean for a point in a vector space to have integer values? Is your vector space a subspace of ${\bf R}^m$ for some $m$? Does "integer values" mean integer coordinates? $\endgroup$ May 29, 2016 at 23:17
  • $\begingroup$ @GerryMyerson: Yes, sorry, a base is fixed and the vertices all have integer coordinates. The lattice elements have integer coordinates, as well. $\endgroup$ May 30, 2016 at 8:47
  • $\begingroup$ Does "regular" mean "all edges have equal lengths"? I ask since regular lattice simplices exist only in certain dimensions. (This is assuming that 1) your simplex is of full dimension and 2) you're using the standard metric.) $\endgroup$ May 31, 2016 at 7:10
  • $\begingroup$ Indeed, I didn't think of that. The simplex arrises in the following way: we take $V$ as the subspace in $\mathbb{R}^{n+1}$ given by $\Sigma x_i = d$. Then the simplex is spanned by vertices $v_i = (a_1, .... a_i - b, ... a_{n+1})$ for appropriate fixed $a_i$ and $b$. So your point might add complications, since it seems that indeed to get what I claim for $S$, one needs to take a non-standard metric. (Anyway perhaps in the end it is not fruitful to view my original problem from this angle, as this seems to be hard) $\endgroup$ Jun 4, 2016 at 10:10

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