Cutting corners. (English) Zbl 0955.52008

First of all in this paper some notions and terms are introduced: corner cut polyhedron \(P_n^d (\subset\mathbb{R}^d)\); \({\mathbb{N}^d \choose n}\) – set of \(n\)-element subsets of \(\mathbb{N}^d\) (with nonnegative integer vectors); staircase; \({\mathbb{N}^d\choose n}_{\text{stair}}\) – set of finite subsets of staircases from \({\mathbb{N}^d\choose n}\); staircase polytope \(Q_n^d (\subset P_n^d)\); corner cut (as specific staircase); \({\mathbb{N}^d\choose n}_{ \text{cut}}\) – set of all \(n\)-element corner cuts.
The authors examine the set \({\mathbb{N}^d \choose n}_{\text{cut}}\) of corner cuts and the corner cut polyhedron \(P_n^d\) from four points of view: polyhedral geometry, computational complexity, enumerative combinatorics and commutative algebra.
Main results are: The corner cut polyhedron satisfies \(P_n^d=Q_n^d +\mathbb{R}^d_{\geq 0}\) and is hence indeed a polyhedron. The staircase polytope \(Q_n^d\) has the same vertex set as \(P_n^d\). The map \(\lambda\to \Sigma\lambda\) defines a bijection between the corner cuts and the common vertex set of \(P_n^d\) and \(Q^d_n\).
There is a polynomial time algorithm for recognizing corner cuts. For fixed \(d\) one obtains \(\#{\mathbb{N}^d \choose n}_{\text{cut}}= O(n^{2d(d-1)/(d+1)})\).
The computational results are translated into algorithms for the Gröbner bases theory: The normal fan of the corner cut polyhedron \(P_n^d\) equals the Gröbner fan of the vanishing ideal of the generic configuration of \(n\) points in affine \(d\)-space.


52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B55 Computational aspects related to convexity
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
05A99 Enumerative combinatorics
14A10 Varieties and morphisms
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