Volume-Distance-Ratio Asymptote

• Volume-Distance-Ratio (VDR) asymptote is a normalized invariant that captures the leading behavior of volume relative to chosen distance scales across various geometric frameworks.
• It distinguishes geometric structures by comparing volume growth to model reference scales, thereby revealing rigidity properties in Hilbert, Riemannian, convex, affine, and Lorentzian settings.
• Analytical methods involve metric comparisons, asymptotic expansions, and differential invariants like the Blaschke metric and Minkowski volume to classify domains and assess inextensibility.

Searching arXiv for the listed papers to ground the synthesis in the cited literature. [Tool call] arXiv search: ids ([1207.1142](/papers/1207.1142)), ([1901.00771](/papers/1901.00771)), ([2405.09379](/papers/2405.09379)), ([1105.6028](/papers/1105.6028)), ([2301.13495](/papers/2301.13495)), ([2508.03263](/papers/2508.03263)), ([1007.2232](/papers/1007.2232)), ([2507.23097](/papers/2507.23097)) The Volume–Distance–Ratio (VDR) asymptote denotes the leading asymptotic behavior of a volume quantity after normalization by a distance scale, typically a power of radius, a model reference volume, or a boundary-normal parameter. In the cited literature, the term appears in several distinct but structurally related settings: metric-ball growth in Hilbert and Riemannian geometries, affine distortion and subset-separation problems in convexity, normalized volume distance near convex hypersurfaces, and timelike-diamond volumes near Lorentzian boundaries. Across these settings, the asymptote functions as a rigidity diagnostic: it distinguishes polytopal from non-polytopal Hilbert domains, detects maximal or linear volume growth under curvature hypotheses, recovers affine invariants such as the Blaschke metric and shape operator, and yields inextensibility criteria for singular spacetimes (Vernicos, 2012, Chen, 2011, Le, 30 Jul 2025).

1. Definitions and normalization schemes

The phrase “VDR asymptote” does not refer to a single normalization used uniformly across all branches of geometry. Rather, the cited works use the same organizing idea—comparing a volume quantity with an appropriate distance scale—to capture leading-order geometry. In Hilbert geometry, the basic object is the metric-ball volume

$V(R):=\Vol_C(B_C(o,R)),$

and the asymptotic volume is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$

These quantities do not depend on the choice of the center $o$. In a 3-dimensional complete non-compact Riemannian manifold with positive scalar curvature forcing at most linear growth, one takes

$\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$

while for expanding gradient Ricci solitons the corresponding quantity is the asymptotic volume ratio

$\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$

provided the limit exists. In Lorentzian geometry, the relevant volume is that of a timelike diamond $I(p,q)$, normalized either by the Minkowski diamond volume or by a fixed multiple of the proper-time scale $\ell^{n+1}$. Near convex hypersurfaces, a natural VDR is the quotient $v(p)/t$, where $v(p)$ is the volume distance and $t$ is affine-normal distance along a centroid line (Vernicos, 2012, Huang et al., 2024, Chen, 2011, Craizer et al., 2010, Le, 30 Jul 2025, Le, 5 Aug 2025).

Setting	Volume quantity	Normalization
Hilbert geometry	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$0	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$1
Riemannian growth	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$2	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$3 or $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$4
Lorentzian causality	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$5	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$6 or $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$7
Affine hypersurfaces	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$8	$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$9

A common source of confusion is the coexistence of large-radius, small-scale, and model-comparison normalizations under the same label. This suggests that the unifying notion is not a fixed formula but a family of leading-order volume-versus-distance invariants.

2. Hilbert geometries: polynomial lower bounds, polytopes, and super-polynomial growth

In an $o$0-dimensional Hilbert geometry $o$1, with Busemann volume $o$2, there exists an explicit constant $o$3, depending only on the dimension $o$4, such that for every point $o$5 and every $o$6,

$o$7

The argument is established more generally for the whole class of proper measures with density, of which the Busemann and Holmes–Thompson volumes are particular cases. Consequently,

$o$8

The decisive rigidity statement is that finite asymptotic volume is equivalent to polytopality: $o$9 Equivalently, precisely the polytopal Hilbert geometries satisfy two-sided bounds

$\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$0

for $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$1 large, so that $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$2. If $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$3 is not a polytope, then $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$4, and

$\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$5

Thus the ratio $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$6 escapes to infinity in every non-polytopal Hilbert domain (Vernicos, 2012).

For convex polytopes the asymptotic coefficient admits a quantitative lower bound in terms of the number $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$7 of vertices: $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$8 for an explicit dimensional constant $\VDR(p)=\limsup_{r\to\infty}\frac{\Vol(B(p,r))}{r},$9. Among $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$0-polytopes, the simplices, having $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$1 vertices, therefore minimize the asymptotic volume within that class. In the special case of the simplex $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$2,

$\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$3

the Euclidean volume of the unit Euclidean ball in $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$4; it is conjectured that $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$5 is the absolute minimum of $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$6 among all Hilbert domains.

The proof architecture combines the Finsler-metric description of the Hilbert norm $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$7, comparison of volume densities via Benzécri compactness, induction on dimension, Crofton and co-area arguments, and a Kreĭn–Milman extremal-point construction. In the non-polytopal case, arbitrarily many distinct extremal boundary points generate disjoint interior balls with centers at almost distance $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$8 from $\AVR(M)=\lim_{r\to\infty}\frac{\Vol(B(O,r))}{r^n},$9, forcing $I(p,q)$0 for any $I(p,q)$1, hence super-polynomial growth relative to degree $I(p,q)$2. The VDR asymptote here is therefore a complete coarse classifier of polytopal versus non-polytopal Hilbert geometry (Vernicos, 2012).

3. Convexity: largest volume ratio and dimension-free distance asymptotics

In asymptotic convex geometry, one VDR-type normalization is encoded by the volume ratio

$I(p,q)$3

and its extremal form

$I(p,q)$4

For every convex body $I(p,q)$5, there is an absolute constant $I(p,q)$6 such that

$I(p,q)$7

This sharp lower bound closes the long-standing gap from the previous best bound of order $I(p,q)$8 down to $I(p,q)$9. Exact $\ell^{n+1}$0-asymptotics are then obtained for several natural classes: if $\ell^{n+1}$1 is the unit ball of a unitary invariant norm on $\ell^{n+1}$2, then $\ell^{n+1}$3; for $\ell^{n+1}$4-Schatten balls $\ell^{n+1}$5, $\ell^{n+1}$6; for full or symmetric tensor products of $\ell^{n+1}$7 endowed with the projective or injective norm, $\ell^{n+1}$8; and if $\ell^{n+1}$9 is unconditional, then $v(p)/t$0 (Galicer et al., 2019).

The lower-bound proof constructs a random polytope

$v(p)/t$1

with $v(p)/t$2 independent uniform samples on $v(p)/t$3 and $v(p)/t$4. A Gluskin-type estimate gives $v(p)/t$5. One then proves that any $v(p)/t$6 sending $v(p)/t$7 into $v(p)/t$8 must satisfy $v(p)/t$9, using a covering-net argument and a small-ball estimate. Klartag’s isomorphic slicing theorem reduces the general case to a body with uniformly bounded isotropic constant. Upper bounds in special classes use Chevet’s inequality, determinant lower bounds for random Gaussian matrices, and symmetry or isotropy through John/Löwner ellipsoids and Bobkov–Nazarov estimates (Galicer et al., 2019).

A different convex-geometric line of work studies the maximal distance between subsets of fixed volume $v(p)$0 inside a unit-volume body: $v(p)$1 For centrally symmetric bounded convex bodies of volume $v(p)$2,

$v(p)$3

where

$v(p)$4

As $v(p)$5,

$v(p)$6

independently of dimension. For the Euclidean ball,

$v(p)$7

For cubes,

$v(p)$8

with upper bound $v(p)$9 and lower bound $t$0. For the regular simplex,

$t$1

and for $t$2-balls with $t$3,

$t$4

All upper bounds are driven by isoperimetric inequalities through the Minkowski–Steiner derivative $t$5 and the profile $t$6 (Ismailov et al., 2023).

These two theories use different objects—affine containment in one case, separation of subsets in the other—but both isolate a leading asymptotic law in which the normalization suppresses raw volume scale and exposes geometric structure. This suggests that, within convexity, VDR asymptotics serve as a coarse invariant of symmetry class and extremal shape.

4. Riemannian manifolds: curvature decay, ends, and asymptotic volume growth

For complete noncompact $t$7-manifolds with asymptotically nonnegative Ricci curvature and a uniformly positive scalar-curvature lower bound, the VDR asymptote is linear. More precisely, let $t$8 satisfy: $t$9 for a continuous nonincreasing $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$00; $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$01 outside a compact set; $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$02; and topological finiteness in the form of $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$03 ends and finite first Betti number. Then

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$04

and

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$05

The constant $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$06 arises from the maximal area of a positively curved $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$07-space in the blow-down limit: after Ricci-flow mollification and pointed Cheeger–Gromov convergence, the limit is a noncollapsed $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$08 space containing a line, hence

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$09

with $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$10 a noncollapsed $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$11 space satisfying $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$12. Convergence of Busemann level sets then bounds each end by at most $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$13, yielding the asymptotic slope $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$14 (Huang et al., 2024).

In the globally nonnegative Ricci case, the Cheeger–Gromoll splitting theorem gives $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$15. If $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$16, then $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$17 isometrically and equality $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$18 occurs. If $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$19, the only equality case is the $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$20-quotient $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$21, with $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$22. In all other cases strict inequality holds. The coarea formula

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$23

and a Sormani-type noncollapsing argument also yield linear lower growth from each end (Huang et al., 2024).

For expanding gradient Ricci solitons $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$24, defined by

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$25

the VDR asymptote is the asymptotic volume ratio

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$26

when the limit exists. If the averaged scalar curvature satisfies

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$27

then

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$28

If moreover

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$29

then there exist $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$30 such that

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$31

Under the faster decay

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$32

there is $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$33 such that

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$34

for every $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$35 and every $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$36, together with the matching upper bound

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$37

If the global $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$38 exists, then

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$39

These local noncollapse estimates, combined with the curvature decay hypothesis, produce pointed $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$40-limits of dilations that are metric cones; if $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$41 is simply connected at infinity, has exactly one end, and $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$42, then every tangent cone at infinity is isometric to $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$43 (Chen, 2011).

Taken together, these results show that the Riemannian VDR asymptote is controlled not only by dimensional scaling but also by end structure, scalar lower bounds, decay of negative Ricci curvature, and sectional-curvature asymptotics.

5. Affine differential geometry: volume distance, Blaschke metric, and shape operator

For a smooth, strictly convex hypersurface $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$44 bounding a convex region, the volume distance from a point $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$45 on the convex side is defined by

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$46

where $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$47 is the hyperplane through $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$48 with unit normal $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$49, $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$50, $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$51 is the enclosed $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$52-dimensional region, and $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$53 is the region between $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$54 and $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$55. At a minimizer $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$56, the criticality condition is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$57

so $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$58 is the centroid of the minimizing section $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$59 (Craizer et al., 2010).

Let

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$60

Then

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$61

and the fundamental dual-Hessian relation is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$62

Thus the Hessian of the volume distance, restricted to the minimizing hyperplane, is directly dual to the normalized second $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$63-derivative of the minimizing volume (Craizer et al., 2010).

Let $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$64, let $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$65 be the affine normal pointing into the convex side, and let

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$66

Under the identification of $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$67 with $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$68, the first asymptotic statement is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$69

where $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$70 is the affine Blaschke metric on $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$71. More precisely,

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$72

where $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$73 and $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$74 is the affine shape operator. Hence the first-order limit recovers the Blaschke metric, while the linear correction recovers the shape operator.

Along the same centroid line, the natural VDR

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$75

admits the expansion

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$76

so that

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$77

With

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$78

this becomes

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$79

The volume-distance asymptote therefore gives a geometric interpretation of both the affine Blaschke metric and the affine shape operator in terms of volume minimization and its Hessian (Craizer et al., 2010).

6. Lorentzian geometry: timelike-diamond asymptotics and spacetime inextensibility

In Lorentzian geometry, the VDR asymptote is defined from the volume of a timelike diamond. For causally related points $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$80 in an $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$81-dimensional globally hyperbolic spacetime $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$82, with proper time

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$83

the diamond is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$84

and its volume is

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$85

The normalization compares this quantity to the corresponding Minkowski diamond volume; one formulation writes

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$86

while another writes the ratio as $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$87 with $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$88 the Lorentzian distance to the boundary. In the flat model used for spatially flat FLRW spacetimes, the Minkowski reference is written as

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$89

and in the inextensibility criteria it is equivalently encoded by a fixed multiple of $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$90 (Le, 5 Aug 2025, Le, 30 Jul 2025).

The central theorem is that any sufficiently regular extension forces the normalized small-diamond asymptote to agree with Minkowski. In a $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$91 or smoother spacetime, one proves that the leading behavior of $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$92 is the Minkowski one up to an $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$93 error, so $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$94. Consequently, if

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$95

there can be no such extension. In the $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$96 criterion, if a finite-length future-directed maximal timelike curve $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$97 satisfies

$\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$98

then no local $\Asvol_n(C):=\liminf_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}, \qquad \overline{\Asvol_n(C)}=\limsup_{R\to\infty}\frac{\Vol_C(B_C(o,R))}{R^n}.$99-extension of $o$00 across $o$01 exists. Under $o$02 regularity, one may take $o$03 to be a timelike geodesic; under $o$04, one may replace $o$05 by affine length $o$06. If $o$07 only, then for any future-ordering sequence $o$08, the condition

$o$09

rules out any continuous extension across $o$10 (Le, 30 Jul 2025).

The criteria admit direct applications. For Christodoulou’s self-similar naked singularity, the volume form in Bondi/self-similar coordinates satisfies

$o$11

and along the timelike geodesic $o$12, $o$13,

$o$14

so $o$15. This implies $o$16-inextensibility at the scaling singularity. For flat FLRW with $o$17, direct integration in conformal time yields

$o$18

for some fixed $o$19, hence $o$20, which rules out any $o$21-extension across the boundary point (Le, 30 Jul 2025).

A more detailed FLRW analysis takes

$o$22

and studies the orthogonal geodesic $o$23. For $o$24, one has

$o$25

and the resulting VDR is strictly smaller than the Minkowski value. For all $o$26, an elementary inequality shows the integrand remains strictly below the $o$27 case, so the same strict inequality persists. For $o$28, the integral

$o$29

can exceed the Minkowski value on a subinterval of $o$30; in particular, for $o$31,

$o$32

Defining

$o$33

one has $o$34 equal to the Minkowski value, $o$35 for small $o$36, and $o$37 for $o$38 sufficiently close to $o$39. By continuity, there exists at least one critical exponent $o$40 with $o$41 equal to the Minkowski value; numerically there is exactly one such $o$42 for each $o$43, lying strictly between $o$44 and $o$45 (Le, 5 Aug 2025).

In this Lorentzian setting, a VDR strictly below the Minkowski constant is interpreted as an “under-volume” of small diamonds and thus stronger geodesic focusing near the boundary, while a VDR strictly above Minkowski indicates “over-volume” and weaker focusing. The critical case, where the first-order asymptote matches Minkowski exactly, is the boundary case in which the VDR criterion alone is inconclusive and higher-order asymptotics or different invariants become necessary (Le, 5 Aug 2025).