On the ball number function. (English) Zbl 1292.28009
It is quite elementary, but nowhere mentioned in most textbooks on geometry or analysis, that for the \(n\)-dimensional Euclidean ball \(B(r)\) with radius \(r>0\) with volume \(V(r)\) and surface content \(O(r)\) the ratios and \(O(r)/nr^{n-1}\) independently of the radius have both the same value \(\pi_n= 2^n\pi^{n/2}/n\Gamma(n/2)\), \(n\geq 2\). The map \(n\to\pi_n\) is the the ball number function for Euclidean balls. The latter function is closely related to \(S(r)\)-adapted formulae for the Lebesgue measure (consequently to the method of indivisibles of Cavalieri and its extension given by Torricelli) and to the thin-layer property of the Lebesgue measure.
The present paper deals with the \(l_{n,p}\)-ball number function \(n\to\pi_n(p)\), which is defined by replacing in the ball number function \(n\to\pi_n\) the Euclidean \(|\cdot|_2\)-norm by the \(|\cdot|_p\)-norm for \(p>1\) and by the \(|\cdot|_p\)-anti-norm for \(0< p< 1\). Again, as for the Euclidean norm, taking now volume and surface of the \(|\cdot|_p\)-balls we get for the corresponding ratios identic values again independent of the radius. The study of the \(l_{n,p}\)-ball number function is motivated by an earlier paper of the author [Lith. Math. J. 49, No. 1, 93–108 (2009; Zbl 1177.60019)], where it was seen, that ball numbers come in by factorizing normalizing constants of density-generating functions.
The paper establishes the following results:
1. \(l_{n,p}\) is a continuous and increasing function with limit \(0\) for \(p\to 0\) and limit \(2^n\) for \(p\to\infty\), \(n\) fixed.
2. There are given asymptotic relations for \(l_{n,p}\) concerning its growth for \(p\to 0\) and \(p\to\infty\).
3. The \(l_{n,p}\)-thin-layer property of the Lebesgue measure.
4. Using an integral representation for the Beta-function there is given a representation formula for the general \(l_{n,p}\)-function reducing it to the Euclidean \(l_{n,2}\).
The paper ends with an outlook concerning more general balls. The latter are meant to be star bodies centered at zero, in particular ellipsoids. Ellipsoid number functions have been studied in an earlier paper of the same author (submitted for publication).
The present paper deals with the \(l_{n,p}\)-ball number function \(n\to\pi_n(p)\), which is defined by replacing in the ball number function \(n\to\pi_n\) the Euclidean \(|\cdot|_2\)-norm by the \(|\cdot|_p\)-norm for \(p>1\) and by the \(|\cdot|_p\)-anti-norm for \(0< p< 1\). Again, as for the Euclidean norm, taking now volume and surface of the \(|\cdot|_p\)-balls we get for the corresponding ratios identic values again independent of the radius. The study of the \(l_{n,p}\)-ball number function is motivated by an earlier paper of the author [Lith. Math. J. 49, No. 1, 93–108 (2009; Zbl 1177.60019)], where it was seen, that ball numbers come in by factorizing normalizing constants of density-generating functions.
The paper establishes the following results:
1. \(l_{n,p}\) is a continuous and increasing function with limit \(0\) for \(p\to 0\) and limit \(2^n\) for \(p\to\infty\), \(n\) fixed.
2. There are given asymptotic relations for \(l_{n,p}\) concerning its growth for \(p\to 0\) and \(p\to\infty\).
3. The \(l_{n,p}\)-thin-layer property of the Lebesgue measure.
4. Using an integral representation for the Beta-function there is given a representation formula for the general \(l_{n,p}\)-function reducing it to the Euclidean \(l_{n,2}\).
The paper ends with an outlook concerning more general balls. The latter are meant to be star bodies centered at zero, in particular ellipsoids. Ellipsoid number functions have been studied in an earlier paper of the same author (submitted for publication).
Reviewer: Werner Strauß (Stuttgart)
MSC:
| 28A50 | Integration and disintegration of measures |
| 28A75 | Length, area, volume, other geometric measure theory |
| 33B15 | Gamma, beta and polygamma functions |
| 53A35 | Non-Euclidean differential geometry |
| 60D05 | Geometric probability and stochastic geometry |
Keywords:
ball number; disintegration of Lebesgue measure; thin-layer property; generalized indivisiblesCitations:
Zbl 1177.60019References:
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