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

This Erdös problem was first stated (for $n = 3$) in [a1], the Klee problem in [a2].