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For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

a lattice 7-gon with no lattice points in the inner polygon http://www.freeimagehosting.net/newuploads/gbl84.png

For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

For $n=5$, this has been shown by Eppstein:
D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28.

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 $[0,1], [1,0], [2,0], [3,2], [3,3], [1,3], [0,2]$.

a lattice 7-gon with no lattice points in the inner polygon http://www.freeimagehosting.net/newuploads/gbl84.png