Publications of (and about) Paul Erdös
Autor: Erdös, Paul; Purdy, G.; Straus, E.G.
Title: On a problem in combinatorial geometry. (In English)
Source: Discrete Math. 40, 45-52 (1982).
Review: Let f(S) be the ratio of the area of a largest (nondegenerate) triangle determined by the points of a finite set S to that of a smallest, and f(n) = infsf(S), where the infimum is taken over all planar, noncollinear sets S of cardinality n. It is known that f(3) = f(4) = 1, and f(5) = (\sqrt5+1)/2; it is clear (by taking S0 to be a set of n points equally spaced and evently distributed on two parallel lines) that f(n) \leq [ ½ (n-1)]. Using the interesting theorem of E. Sas that the ratio \rho of the area of a convex set C to that of a triangle contained in C having maximal area satisfies the inequality \rho \leq 4\pi/3\sqrt3 < 2.4184, the authors prove that f(n) = [(n-1)/2] for n > 37, and that, moreover, for even n \geq 38, if f(S) = f(n) then S is affinely equivalent to the set S0 mentioned above. It is conjectured that f(n) = [(n-1)/2] also for 5 < n \leq 37, but in this range other extremal configurations besides S0 are possible. Several other excellent unsolved problems are stated.
Classif.: * 52A40 Geometric inequalities, etc. (convex geometry)
Keywords: discrete geometry; ratio of areas; largest triangle; smallest triangle
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