The convex hull of the lattice points inside a curve

Huxley, Martin Neil 2014. The convex hull of the lattice points inside a curve. Periodica Mathematica Hungarica 68 (1) , pp. 100-118. 10.1007/s10998-014-0024-5

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Abstract

Let C be a smooth convex closed plane curve. The C -ovals C(R,u,v) are formed by expanding by a factor R , then translating by (u,v) . The number of vertices V(R,u,v) of the convex hull of the integer points within or on C(R,u,v) has order R 2/3 (Balog and Bárány) and has average size BR 2/3 as R varies (Balog and Deshouillers). We find the space average of V(R,u,v) over vectors (u,v) to be BR 2/3 with an explicit coefficient B , and show that the average over R has the same B . The proof involves counting edges and finding the domain D(q,a) of an integer vector, the set of (u,v) for which the convex hull has a directed edge parallel to (q,a) . The resulting sum over bases of the integer lattice is approximated by a triple integral.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Publisher:	Springer
ISSN:	0031-5303
Last Modified:	05 Mar 2019 14:25
URI:	https://orca.cardiff.ac.uk/id/eprint/58596

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