Hadwiger—Wills-Type Higher-Dimensional Generalizations of Pick's Theorem

Kołodziejczyk, Krzysztof

doi:10.1007/s004540010041

Cited by 4 publications

(2 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 2.1 is the two-dimensional case of the higher dimensional volume formulas of [2] in terms of weight functions; and all the volume formulas of [13,14,17,18] can be induced from those volume formulas of [2] by choosing weight equal to 1. The volume formula of [12] is also a special case of the volume formula of [2] by setting weight equal to 1.…”

Section: Weighted Version Of the Pick Theoremmentioning

confidence: 99%

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Chen

2003

Annals of Combinatorics

View full text Add to dashboard Cite

This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron, then using the idea of weight function of [2] to obtain a weighted version; other Pick type theorems known to the author for integral lattice Z 2 are reduced to some special cases of the generalization. Finally, using the idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions. Point-Set Version of the Pick TheoremLet P be a lattice polygon of R 2 , i.e., the vertices of P are points of the integral lattice Z 2 . Let i(P ) be the number of lattice points of P and i(∂P ) the number of lattice points of its boundary ∂σ. The Pick Theorem says thatGiven a bounded lattice polyhedron X of R 2 ; we denote byX the closure of X and by int X the interior of X; the frontier of X is the setX − intX. The link of X near a point x ∈X is the intersection of X and a circle S 1 (x, r) centered at x with small enough radius r; the Euler characteristic χ(lk (x, X)) is a local topological invariant, which plays an important role in our Pick type theorems. For an interior point x ∈ int X, the link lk (x, X) is a circle and has Euler characteristic zero. If X is closed, then for any x ∈ fr X, the link lk (x, X) is a collection of finite number of arcs and points, so χ(lk (x, X)) = the number of branches near x.

show abstract

Section: Weighted Version Of the Pick Theoremmentioning

confidence: 99%

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Chen

2003

Annals of Combinatorics

View full text Add to dashboard Cite

show abstract

“…Since 1960, many papers have been published concerning Pick's formula. They contain several proofs of the formula [1,3,4,11,12,16,20,27,28,29] or proof of the equivalence of this result with other ones [5,7,8,15] or generalizations to more general polygons [2,6,9,10,24,25,26,30] to more general lattices [19,23] and also to higher-dimensional polyhedra [13,21,22,25].…”

Section: Introductionmentioning

confidence: 99%

Euler's characteristics and Pick's theorem

Dubeau¹,

Labbé²

2007

ijcms

View full text Add to dashboard Cite

Pick's Theorem is used to compute the area of lattice polygons. In this paper we present a very general definition of a polygon to obtain the definition of faces and holes of a polygon. The decomposition of a polygon into triangles is briefly discussed. Then lattices are introduced with their corresponding elementary triangles and triangulations. We recall and prove Pick's Theorem for simple polygons. Then a generalization of Pick's Theorem is proved for very general lattice polygons, generalized lattice polygons and union of generalized lattice polygons using original and modified Euler's characteristics.

show abstract

Elementary geometry on the integer lattice

Maehara¹,

Martini

2018

Aequat. Math.

View full text Add to dashboard Cite

Hadwiger—Wills-Type Higher-Dimensional Generalizations of Pick's Theorem

Cited by 4 publications

References 10 publications

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

Euler's characteristics and Pick's theorem

Elementary geometry on the integer lattice

Contact Info

Product

Resources

About