TY - JOUR

T1 - Points covered an odd number of times by translates

AU - Pinchasi, Rom

N1 - Publisher Copyright: © The Mathmatical Association of America.

PY - 2014/8/1

Y1 - 2014/8/1

N2 - Given an odd number of axis-aligned unit squares in the plane, it is known that the area of the set whose points in the plane that belong to an odd number of unit squares cannot exceed the area of one unit square, that is, 1. In this paper, we consider the same problem for other shapes. Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T. This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.

AB - Given an odd number of axis-aligned unit squares in the plane, it is known that the area of the set whose points in the plane that belong to an odd number of unit squares cannot exceed the area of one unit square, that is, 1. In this paper, we consider the same problem for other shapes. Let T be a fixed triangle and consider an odd number of translated copies of T in the plane. We show that the set of points in the plane that belong to an odd number of triangles has an area of at least half of the area of T. This result is best possible. We resolve also the more general case of a trapezoid and discuss related problems.

UR - http://www.scopus.com/inward/record.url?scp=84954451269&partnerID=8YFLogxK

U2 - https://doi.org/10.4169/amer.math.monthly.121.07.632

DO - https://doi.org/10.4169/amer.math.monthly.121.07.632

M3 - مقالة

SN - 0002-9890

VL - 121

SP - 632

EP - 636

JO - American Mathematical Monthly

JF - American Mathematical Monthly

IS - 7

ER -