Some external problems in geometry

✍ Scribed by George Purdy

• Publisher: Elsevier Science
• Year: 1974
• Tongue: English
• Weight: 805 KB
• Volume: 7
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(74)90041-7

✦ Synopsis

The question of how uften the same dzstance can occur between k distinct points in n-dimensional Euclidean space E, ;~a: been exteasively studied by Paul Erdijs and others. Sir Alexander Oppenheim posed the somewhat simi:ar problem of investigating how many triangle:; with vertices chosen from ar$Io.I,J k pints in E, :an have the same non-zero area. A subsequent article by Erdds and Purdy gave some preliminnrir results on this problem. Here we carry that work somewhat further and show ;har: there can-lot be more than ck3' ' triangles with the same non-zero area chosen from among k points in ES, where E is a positive constant. Since there can bq: ck3 such triangles in Es, the result i, in a certain sense best possible. The methods used are mainly combinatorial and geometrical. .4 wb,n tool is .a theorem on generalizzd graphs due to Paul Erdbs.