Volume-Regularity of Distance-Induced Balls

• Volume-regularity of distance-induced balls is defined by uniform control over the measure growth of generalized h-balls at small scales.
• It underpins key results in comparison geometry and geometric flows, linking ball volume bounds to topological collapse and Uryson width constraints.
• This concept extends classical Ahlfors regularity to non-homogeneous distance functions, with applications in statistical identifiability and metric analysis.

Volume-regularity of distance-induced balls encompasses analytic and geometric conditions on the measure growth and regularity properties of balls defined via a generalized distance function on a metric measure space. These properties play a central role in comparison geometry, limit space analysis in geometric flows, statistical identifiability via distance distributions, and topological width-type collapse phenomena. Contemporary research establishes precise equivalences between volume-regularity, topological collapse, and analytic regularity, and extends classical results to broad classes of non-homogeneous, non-translation-invariant distances.

1. Formal Definitions and Distinctions

Let $(X,h,\mu)$ be a metric measure space, where $h:X\times X\to[0,\infty)$ is a generalized (not necessarily symmetric nor translation-invariant) distance, and $\mu$ is a Borel measure. For each $x\in X$ and $t>0$, define the $h$-ball $B_t^{(h)}(x) = \{ y\in X : h(x,y)<t \}$, with volume function $\Phi(x,t) := \mu(B_t^{(h)}(x))$.

A distance $h$ is volume-regular (as formalized in (Betken et al., 6 Nov 2025)) if there exists $\varepsilon>0$ so that for almost every $x$, the following hold:

• $\lim_{t\to 0^+} \Phi(x,t) = 0$;
• for all $0<t<\varepsilon$, $0<\Phi(x,t)<\infty$;
• there exists a reference point $y$ and functions $\delta_t(x,y)=\Phi(x,t)/\Phi(y,t)$, $\delta(x,y)$, and constants $c,C>0$, such that $$c\,\delta(x,y)\leq \delta_t(x,y)\leq C\,\delta(x,y)$$ for $0<t<\varepsilon$.

This definition encompasses, but is not limited to, the standard Ahlfors $\alpha$-regularity condition, which requires that $c\,t^\alpha\leq \mu(B_t^{(h)}(x))\leq C\,t^\alpha$ for all $x$ and $0<t<\varepsilon$. In this setting, $\delta(x,y)\equiv1$, and volume-regularity reduces to uniform comparability of ball volumes at small scales.

2. Volume-Regularity in Riemannian and Metric Geometries

Volume-regularity of distance-induced balls captures growth upper bounds crucial for geometric analysis on Riemannian manifolds and metric spaces.

A fundamental result, originally due to Guth and reproved by Papasoglu (Papasoglu, 2019), states:

There exists $E_n>0$ such that if a closed Riemannian manifold $(M^n,g)$ satisfies $\operatorname{Vol}_g(B(x,r))\leq E_n r^n$ for all $x$ and $r\leq R$, then the $(n-1)$-Uryson width of $(M,g)$ is at most $R$: $\mathrm{UW}_{n-1}(M,g)\leq R$.

The same theorem holds verbatim for compact metric spaces with $\operatorname{HC}_{n}(B(x,r)) \leq E_n r^n$, where $\operatorname{HC}_n$ is the $n$-dimensional Hausdorff content (Papasoglu, 2019).

A refinement shows that it suffices to control the $(n-1)$-content of a neighborhood of the boundary of each $R$-ball, rather than the full $n$-volume---the codimension-1 content condition: if for every $x$, $B(x,R)$ is contained in some open $U\subset B(x,10R)$ with $$\operatorname{HC}_{n-1}(\overline U) \leq E'_n R^{n-1}$$, then $\mathrm{UW}_{n-1}(X)\leq R$ for the space $X$ (Papasoglu, 2019). This demonstrates that volume-regularity at codimension governs topological collapse phenomena.

Definition/Condition	Formulation	Context
Volume-regularity	See definition in Section 1	(Betken et al., 6 Nov 2025)
Ahlfors $\alpha$-regular	$c\, t^\alpha\leq \mu(B_t(x))\leq C\,t^\alpha$	(Betken et al., 6 Nov 2025), any measure space
Uryson width	Existence of map to $q$-complex with fibers $\leq W$	(Papasoglu, 2019)
Hausdorff content	$\inf\{\sum_i r_i^n: U\subset\bigcup_i B(x_i,r_i)\}$	(Papasoglu, 2019)

3. Volume Comparison, Regularity, and Continuity of Volume Functions

The classical Bishop–Gromov comparison theorem establishes that, under Ricci curvature lower bounds, the ratio $$r\mapsto \operatorname{Vol}_g(B_p(r))/\operatorname{Vol}_\kappa(B^\kappa(r))$$ is non-increasing, where $B^\kappa(r)$ is a ball of radius $r$ in the simply-connected space of constant curvature $\kappa$ (Graf, 2015). Graf proves that this monotonicity and the associated volume-growth and diameter bounds persist under merely $C^{1,1}$ regularity of the metric and $L^\infty$ Ricci curvature lower bounds.

Regularity of the volume function $r\mapsto \operatorname{Vol}_g(B_p(r))$ is established as:

• continuous for all $r>0$,
• absolutely continuous in $r$,
• a.e. differentiable with $$d/dr\,\operatorname{Vol}_g(B_p(r))=\operatorname{Area}_g(S_p(r))$$ for almost every $r$.

This persists in the $C^{1,1}$ regime, as does the validity of Myers’ diameter bound and the finiteness-of-diameter results relevant to singularity theorems (Graf, 2015).

Explicit calculations for model spaces, such as homogeneous lens spaces, provide closed-form and piecewise real-analytic expressions for the geodesic ball volume function $V_n(r)$ (see Section 3 of (Balch et al., 2020)). For example, on $L(n;1)=S^3/\mathbb Z_n$,

$$V_n(r)=\begin{cases} 2\pi(r-\sin r \cos r), & 0\leq r\leq \pi/n,\ \frac{2\pi^2}{n}-2\pi\cos^2 r\tan(\pi/n), & \pi/n\leq r\leq \pi/2, \end{cases}$$

with regularity $V_n\in C^1$ but not $C^2$ at $r=\pi/n$ (Balch et al., 2020).

4. Volume-Regularity under Geometric Flows and Analytical Constraints

In Ricci flow backgrounds, volume-regularity admits scale-uniform two-sided control even without lower bounds on Ricci curvature (Zhang, 2011). The $\kappa$ non-inflating property guarantees, under scalar curvature upper bound $R\leq \alpha/(t_0-t)$ on parabolic cubes, scale-invariant upper bounds $|B_{g(t_0)}(x_0,r)|\leq \kappa r^n$. Coupling this with Perelman’s non-collapsing lower bounds yields

$$\kappa_{nc}\,r^n\leq |B_{g(t)}(x,r)|\leq \kappa_{infl} r^n,$$

implying a uniform volume-doubling property

$|B_{g(t)}(x,r)|\leq D\,|B_{g(t)}(x,r/2)|,$

where $D$ depends on geometric and flow parameters (Zhang, 2011). No Ricci curvature lower bound is required; this property is critical for PDE techniques in Ricci flow and metric measure geometry.

In the context of scalar curvature and entropy lower bounds, together with almost-Euclidean upper volume growth, $\varepsilon$-regularity theorems guarantee that unit balls are Gromov–Hausdorff close, and even bi-Hölder and $W^{1,p}$ homeomorphic, to Euclidean balls (Neumayer, 2022). This regularity propagates to limit spaces in measured Gromov–Hausdorff topology, establishing rectifiability and strong metric regularity solely from volume-regularity constraints.

5. Volume-Regularity in Distance-based Statistical Identifiability

Volume-regularity is a central analytic condition in the extension of interpoint distance-based characterizations of probability laws (Betken et al., 6 Nov 2025). The identifiability theorem holds for any volume-regular generalized distance $h$ on $\mathbb R^k$: if two independent samples $X_i\sim f$, $Y_j\sim g$ satisfy $$h(X_1,X_2)\stackrel{\mathcal D}{=}h(Y_1,Y_2)\stackrel{\mathcal D}{=}h(X_3,Y_3)$$, then $f=g$ (subject to Lebesgue differentiability, bounded oscillation, and integrability conditions also depending on the small-scale ratios $\delta_t(x,y)$ given in volume-regularity).

Quantitative stability is established: under uniform bounds $0<\delta_*<\delta(x,\xi)<\delta^*<\infty$ and for small $t$,

$$\|f-g\|_{L^2}^2 \leq \frac{1}{c\,\delta_*}\left[\frac{\Delta_K(t)}{\Phi(\xi,t)}+r(\xi,t)\right].$$

Here, $\Delta_K(t)$ is a Kolmogorov-type discrepancy between interpoint distance distributions, and $\Phi(\xi,t)$ is the $h$-ball volume at the reference point (Betken et al., 6 Nov 2025).

Ahlfors $\alpha$-regularity (as in many classical settings) implies volume-regularity: $\delta(x,y)\equiv1$ and all ball volumes are uniformly comparable.

6. Illustrative Examples and Analytic Structures

Volume-regularity covers a wide spectrum of distances beyond the Euclidean, and includes:

• Canberra distance: $h(x,y)=\sum_{i=1}^k |x_i-y_i|/(|x_i|+|y_i|)$. In $k=1$, $\Phi(x,t) = \frac{4t|x|}{1-t^2}$ for $x\neq0$ and $0
• Entropic (Kullback–Leibler type) distances: $h(x,y)=\sum_i |x_i\log(x_i/y_i)-x_i+y_i|$. For $k=1$, ball volumes scale as $\Phi(x,t)\asymp t^{1/2}$.
• Bray–Curtis dissimilarity: $$h_{BC}(x,y)=\sum_i\omega_i(x,y) |x_i-y_i|/(x_i+y_i)$$ with strictly positive weights $\omega_i$; volume comparisons reduce to those for the Canberra metric.

Volume-regularity is satisfied for geodesic balls on smooth compact Riemannian manifolds, as a consequence of the classical asymptotic expansions and measure comparison estimates (cf. (Betken et al., 6 Nov 2025, Graf, 2015)). In all these settings, analytic controls on small ball volumes underpin identifiability theorems, limit structures in metric geometry, and continuity properties for statistical and geometric functionals.

7. Broader Significance and Interplay with Topology and Analysis

Volume-regularity of distance-induced balls forms a quantitative bridge between metric measure geometry, analysis, and probability. Uniform upper volume-growth of balls translates directly into topological collapse (low Uryson width), analytic regularity of function spaces (doubling and Poincaré inequalities), and statistical identifiability of probability laws via distance statistics. Volume-regularity, rather than strict homogeneity or translation invariance, is now established as the critical, unifying analytic hypothesis for extension of classical results across geometry and statistics.

A plausible implication is that further weakening of ball-volume growth constraints---for example, by relaxing upper bounds to hold only on boundaries or on non-dense subsets---could yield even broader collapse and regularity results, although counterexamples suggest that controlling boundary content is the minimal viable condition (Papasoglu, 2019). The development of this analytic paradigm promises deeper connections between geometric analysis, topological dimension theory, and high-dimensional statistical learning.