From: Bill Dubuque <wgd@nestle.ai.mit.edu>
Subject: Pick's formula in higher dimensions [was: Russian math olympiad problem on lattice]
Date: 24 Aug 2000 02:05:11 -0400
Newsgroups: sci.math
Summary: [missing]

Hull Loss Incident <x@y.z.com> wrote:
>
> > |Pick's theorem is that the area of a lattice polygon
> > |without holes is  #points inside + 1/2(#points on boundary) - 1.
> >
> also add that the formula can be extended to polygons with multiple
> holes and connected components, but not in an obvious way to 3-dim
> For a tetrahedron the volume is not determined by the number of  
> lattice points in the interior, faces, vertices and edges.

There are some beautiful higher-dimensional extensions of Pick's formula
based upon recent deep work in combinatorial algebraic geometry, in
particular around toric varieties, with contributions by Brion, Cappell,
Khovanskii, Morelli, Pommersheim, Shaneson, etc.  For a readable
introduction see Morelli's paper [1], and [2] for an online start.

-Bill Dubuque

[1] Morelli, Robert.  Pick's theorem and the Todd class of a toric variety. 
Adv. Math. 100 (1993), no. 2, 183--231.  MR 94j:14048

[2] http://www.emis.math.ca/EMIS/journals/ERA-AMS/1996-01-001/1996-01-001.tex.html