On the volume of a hyperbolic simplex

Gausz uses the word ‘jungle’ in relation to volumes in non-Eucidean spaces, cf. Fejes Tóth [4], p. 301. This may refer to the highly complicated transcendental methods involved. The present paper states some facts and some speculations which point at a different direction. However, we do not present explicit solutions for existing problems. The basic fact is that the n points of any hyperbolic (n— 1)-simplex are uniquely represented by a nonnegative doubly stochastic symmetric matrix of order n. This suggests a function of this matrix as a more algebraic expression for the volume of a hyperbolic simplex. The resemblance of recent results on simplices with maxi mum volume, and on the van der Waerden conjecture suggest a role for the permanent in the definition of volume. This idea is supported by certain connections between exterior and symmetric algebra. Perhaps there are relations with Poincaré’s formula. We do not present a coherent theory of volume. But maybe our facts and speculations contribute some material (of an algebraic and combinatorial nature) for the construction of a passable road through the jungle mentioned above.