On the ball number function

Abstract

The ball number function was recently defined in [W.-D. Richter, Continuous l _n,p -symmetric distributions, Lith. Math. J., 49(1):93–108, 2009] for l _n,p -balls. It was shown there that ball numbers occur naturally in factorizations of the normalizing constants of density-generating functions if such functions depend on the l _n,p -norm. For an analogous situation in the case of ellipsoids, we refer to [W.-D. Richter, Ellipses numbers and geometric measure representations, J. Appl. Anal., 17, 2011 (in press)] and [W.-D. Richter, Geometric and stochastic representations for elliptically contoured distributions (submitted for publication)]. Here, we discuss some additional properties of the ball number function and state the problem of extending it to sets generated by arbitrary norms or anti-norms (for definition, see [M. Moszyńska and W.-D. Richter, Reverse triangle inequality. Anti-norms and semi-anti-norms (submitted for publication)]), and even to more general balls. As an application, we present a method for deriving new specific representation formulae for values of the Beta function.