Erdös problem

The problem on the existence in an $ n $- dimensional Euclidean space $ E ^ {n} $ of a set of more than $ 2 ^ {n} $ 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 $ 2 ^ {n} $ elements if and only if it consists of the set of vertices of a rectangular parallelopipedon in $ E ^ {n} $. The proof of this assertion also solved the so-called Klee problem: What is the number of vertices $ m ( K) $ of a polyhedron $ K \subset E ^ {n} $ if any two of its vertices lie in distinct parallel supporting hyperplanes of $ K $( the Klee property). If a set $ N \subset E ^ {n} $ has the Erdös property, then the convex hull $ M = \mathop{\rm conv} N $ of $ N $ is a polyhedron having the Klee property and $ m ( M) $ is equal to the cardinality of $ N $. If a polyhedron $ K $ has the Klee property, then $ m ( K) \leq 2 ^ {n} $. The equality $ m ( K) = 2 ^ {n} $ characterizes $ n $- dimensional parallelopipeda in the set of all polyhedra having the Klee property.

The Erdös problem is connected with the Hadwiger hypothesis $ b ( M) = m ( M) $.

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 $ n = 3 $) 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)