# On lattice points "far inside" convex lattice polygons

Let $\mathcal{P}$ be a convex lattice polygon with $n$ vertices and let $\mathcal{L}$ be the set of all lattice points inside $\mathcal{P}$. For every $n \geq 5$, does there exist a point in $\mathcal{L}$ such that it also lies in the convex polygon bounded by (all) the diagonals of $\mathcal{P}$? How many such points are there? (//By diagonals I mean of course the lines different from the sidelines of the polygon which are connecting two vertices of $\mathcal{P}$.)

I proved a while ago that for $n=5$ there is such a point in $\mathcal{L}$. I also managed to show this now for $n \geq 6$ using a similar argument, yet it got more involved and I still need to check for potential bugs. Any ideas for the general case?

• I suspect there are simple counterexamples for n=6, so I may be misunderstanding something. Can you say more about what interior region is supposed to have a lattice point? Gerhard "Ask Me About System Design" Paseman, 2011.11.17 Commented Nov 17, 2011 at 10:07
• What do you mean by "the convex pentagon bounded by (all) the diagonals of P"? usually the do not bound a pentagon. Commented Nov 17, 2011 at 11:41
• Yes he does, in the first line. Commented Nov 17, 2011 at 13:10
• @Fedor: Sorry, it should have read "the convex polygon bounded by all the diagonals of $\mathcal{P}$. Commented Nov 17, 2011 at 15:23
• I am still having difficulty understanding the phrase, "the convex polygon bounded by all the diagonals of $P$." In general, there is no convex polygon bounded by all the diagonals, if by "bounded" you mean, "forming the boundary of." There are many convex polygons, each bounded by a subset of the diagonals... Commented Nov 17, 2011 at 17:56

For $$n=5$$, this has been shown by Eppstein:
For odd $$n>5$$, one could consider the polygon bounded by the longest diagonals.
It may be defined as the intersection of the half-planes containing $$(n+1)/2$$ vertices of $$P$$.
For $$n=9$$, this intersection may be empty (for example, if the nine vertices form three triples and each of the triples is placed very close to a vertex of a regular triangle).
For $$n=7$$, this intersection is non-empty. However, it may be free of lattice points: take, for example, the polygon $$P$$ with vertices $$[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$$.