Model space radial area and ball volume

Assume the inherited Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the polar surface-measure supplier. Let $n\ge 2$ and $k\in \mathbb{R}$, and let ${\omega }_{n-1}$ be the total surface measure of the unit sphere $S^{n-1}\subseteq {\mathbb{R}}^n$, that is, its polar surface measure The polar surface set function on the unit sphere, whose finite Borel-measure property is supplied by Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma. The model radial area function is $A_k(r):={\omega }_{n-1}{\mathrm{sn}}_k(r{)}^{n-1},$ defined on the positive domain of the comparison sine, for $0<r<\pi \mathrm{/}\sqrt{k}$ when $k>0$ and for every $r>0$ when $k\le 0$. Set $A_k(0):=0$; when $k>0$ also set $A_k(\pi \mathrm{/}\sqrt{k}):=0$, the continuous endpoint value. Thus for $k\le 0$ the formula extends to all $r\ge 0$ and is positive for $r>0$.

The model ball volume function is the integral $V_k(r):={\int }_0^r{A}_k(t)dt,$ for $0\le r\le \pi \mathrm{/}\sqrt{k}$ when $k>0$ and for all $r\ge 0$ when $k\le 0$. The endpoint integral exists because $A_k$ extends continuously there. It is computed with the one-dimensional Lebesgue integral of the continuous nonnegative function $A_k$ (The polar surface set function on the unit sphere); the value at the left endpoint is $V_k(0)=0$. When $k<0$ the substituted formulas $A_k(r)={\omega }_{n-1}{\sinh }^{n-1}(\sqrt{-k}r)\mathrm{/}(\sqrt{-k}{)}^{n-1}$ and $V_k(r)={\omega }_{n-1}{\sqrt{-k}}^{1-n}{\int }_0^r{\sinh }^{n-1}(\sqrt{-k}t)dt$ and the spherical formulas for $k>0$ express these functions through the displayed comparison-sine data only.

The saturated model volume is $V_k^{\star }(r):=\{\begin{array}{ll}{V}_k(\min \{r,\pi \mathrm{/}\sqrt{k}\}),& k>0,\\ V_k(r),& k\le 0,\end{array}$ for every $r\ge 0$. Thus for $k>0$ the saturated volume is constant for $r\ge \pi \mathrm{/}\sqrt{k}$, at the value $V_k(\pi \mathrm{/}\sqrt{k})$ of the whole model sphere configuration, while for $k\le 0$ no saturation occurs. The saturation is a convention of this page, used by Bishop–Gromov comparison, and it is stated here once and for all.

These functions are the radial area and the volume of the model space of constant curvature $k$; that geometric identification is proved by the comparison items of this page together with the model-space examples, and is not part of this definition. Only the displayed formulas, their domains, the positivity of ${\mathrm{sn}}_k$ on the positive domain, and the saturation convention are asserted here. For $n=2$ the exponent is $1$, so $A_k(r)={\omega }_1{\mathrm{sn}}_k(r)=2\pi {\mathrm{sn}}_k(r)$; in the zero-dimensional case $n=0$ no spherical factor and no radial formula occur, and the notation is not used.