Erdös problem

The problem on the existence in an -dimensional Euclidean space of a set of more than points, any three of which form a non-obtuse triangle (the Erdös property). It was posed by P. Erdös (see [1]), who also made the conjecture (proved in [2]) that the problem has a negative answer and that a set having the Erdös property contains elements if and only if it consists of the set of vertices of a rectangular parallelopipedon in . The proof of this assertion also solved the so-called Klee problem: What is the number of vertices of a polyhedron if any two of its vertices lie in distinct parallel supporting hyperplanes of (the Klee property). If a set has the Erdös property, then the convex hull of is a polyhedron having the Klee property and is equal to the cardinality of . If a polyhedron has the Klee property, then . The equality characterizes -dimensional parallelopipeda in the set of all polyhedra having the Klee property.

The Erdös problem is connected with the Hadwiger hypothesis .

References

[1]	P. Erdös, "Some unsolved problems" Michigan J. Math. , 4 (1957) pp. 291–300
[2]	L. Danzer, B. Grünbaum, "Ueber zwei Probleme bezüglich konvexer Körpern von P. Erdös und von V.L. Klee" Math.Z. , 79 (1962) pp. 95–99

Comments

This Erdös problem was first stated (for ) in [a1], the Klee problem in [a2].

References

[a1]	P. Erdös, "Problem 4306" Amer. Math. Monthly , 55 (1948) pp. 431
[a2]	V.L. Klee, "Unsolved problems in intuitive geometry" , Seattle (1960) (Mimeographed notes)