Maximal lattice-free convex bodies introduced by Herb Scarf and the related complex of maximal lattice free simplices (also known as the Scarf complex) are remarkable geometric constructions with deep connections to combinatorics, convex geometry, integer programming, game theory, fixed point computations, algebraic geometry, and economics. It is certainly an object I would like to think more and tell you more about. Here is a sculpture made by artist Ann Lehman based on such a body generated by a certain 4 by 3 matrix.

**Ann Lehman**: A sculpture based on a maximal lattice free convex body (photograph taken by the sculptress).

The body described in this sculpture is topologically a (two dimensional) disk, triangulated in a specific way. The boundary is a polygon embedded in 3-space consist of 21 “blue” edges. The “black” edges are internal edges of the triangulation. The triangles of the triangulation are not part of the sculpture but it is easy to figure out what they are and the shape has a remarkable zigzag nature. All vertices are integral. The only interior integral point is in “red”. (The “green” point is the origin, it is *not* in the body.)

Herb Scarf, me , and the sculpture (picture taken by Kareen Rozen)

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Perhaps it would be nice if you elaborate on some of its deep connections to the fields you mentioned.

Rafee Kamouna.

Some related post is here https://gilkalai.wordpress.com/2019/03/22/danny-nguyen-and-igor-pak-presburger-arithmetic-problem-solved/ . It would be great (but not easy) to have a post explaining Scarf’s theory, and its origin in Scarf’s famous results on cores of games without side payments, his economics theory of computing equilibria of games, and his subsequent works related to integer programming.

It would also be nice to see the planar realization of the body, with the points at their correct integer coordinates.

ロットリング 万年筆

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