Distinct Triangle Areas in a Planar Point Set
نویسندگان
چکیده
Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least b 2 c, which is attained for dn/2e and respectively bn/2c equally spaced points lying on two parallel lines. We show that this number is at least 17 38 n − O(1) ≈ 0.4473n. The best previous bound, ( √ 2 − 1)n − O(1) ≈ 0.4142n, which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar’s theorem [23] on the number of distinct directions determined by n noncollinear points in the plane.
منابع مشابه
Areas of triangles and Beck's theorem in planes over finite fields
Abstract. In this paper, we will show that if E is a subset of a planar point set over a finite field Fq (q not necessarily prime) and has cardinality |E| > 64q log2 q, then there are more than q 2 distinct areas of triangles sharing the same vertex. A finite field version of Beck’s theorem is derived to prove the result. The theorem says that if |E| > 64q log2 q, then pairs of distinct points ...
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