Head-Tail Diameter Bound

• Head-tail diameter bound is a framework defining upper (head) and asymptotic (tail) estimates for the diameter of convex structures and manifolds.
• It leverages recursive methods like the Kalai–Kleitman approach to derive quasipolynomial and polynomial tail bounds in high-dimensional, large-facet regimes.
• In Ricci flow, analogous bounds provide explicit estimates based on volume, curvature, and geometric invariants, bridging discrete and continuous frameworks.

The head-tail diameter bound comprises a collection of upper and lower bounds for the diameter of certain mathematical structures, most prominently convex polyhedra, polytopes, and Riemannian manifolds evolving under geometric flows. "Head" bounds refer to upper estimates, while "tail" bounds—also called "tail bounds" in polyhedral combinatorics—refer to asymptotic upper estimates that hold once structural parameters pass regime-specific thresholds. In Ricci flow, "tail" bounds may describe explicit lower limits. The sharpness, formulation, and proof techniques of these bounds reflect advancements in combinatorics, geometry, and analysis.

1. Fundamental Definitions

For a $d$-dimensional convex polyhedron $P \subset \mathbb{R}^n$ with $n$ facets, the graph diameter $\operatorname{diam}(P)$ is defined as the maximum, over all pairs of vertices $u, v$ of $P$, of the shortest-path distance between $u$ and $v$ in the $1$-skeleton of $P$ (the edge graph). Polytopes are bounded polyhedra; normal simplicial complexes and simplicial pseudomanifolds generalize combinatorial structures whose dual graphs also admit diameter notions.

A tail bound is an upper bound on $\operatorname{diam}(P)$ that holds once the number of facets $n$ exceeds a certain function of the dimension $d$ (typically denoted $f(d)$). Tail-quasipolynomial bounds have the form $\operatorname{diam}(P) = O((n-d)^{O(\log d)})$, while tail-polynomial bounds have the form $\operatorname{diam}(P) = O(n^k)$ or $O(n^{1+\epsilon})$ for $n$ past a threshold. For Ricci flow on compact manifolds $\left(M^n, g(t)\right)$, the head-tail bounds refer to explicit upper and lower estimates on $\operatorname{diam}(M, g(t))$ in terms of volume, scalar curvature, and geometric invariants.

2. Key Results for Polyhedra and Polytopes

The study of head-tail bounds in polyhedral combinatorics began with the Kalai–Kleitman recursive approach, which led to the first quasipolynomial diameter bounds for convex polyhedra:

• Kalai–Kleitman recursion (1992): For $d > 2$, $n \ge d$,

$$\Delta(d, n) \leq \Delta(d-1, n-1) + 2 \Delta\big(d, \lfloor n/2 \rfloor\big) + 2$$

where $\Delta(d, n)$ denotes the maximum graph diameter of any $d$-polyhedron with $n$ facets.

• Tail-quasipolynomial bounds:
  • For convex polyhedra, for $d > 3$ and $n > d$,

  $\mathrm{A}_u(d,n) \leq (n-d)^{\log_2(d-1)}$

  with the sharper bound

  $\mathrm{A}_u(d,n) \leq (n-d-1)^{\log_2(d-1)}$

  for $n \ge 2d$. - For convex polytopes, for $d \ge 3$ and $n > d$,

  $$\mathrm{A}_b(d, n) \leq \left\lceil \frac{2}{3}(n-d+\frac{3}{2}) \right\rceil^{\log_2(d-1)}$$

  and for $n \ge 2d$,

  $$\mathrm{A}_b(d, n) \leq \left(\frac{2}{3}(n-d+\frac{3}{2})\right)^{\log_2(d-1)}$$ - For pure, normal $(d-1)$-dimensional simplicial pseudomanifolds with $n$ vertices,

  $E(d, n) \leq (n-d)^{\log_2 d},\quad d \ge 4,~n > d$

• Tail-polynomial and almost-linear bounds (via iterations and binomial coefficient estimates):
  • For polyhedra, a tail-cubic bound

  $\Delta(d, n) \leq (n - d)^3$

  holds for $n-d \ge 2^{d-3}$, and the tail-almost-linear bound

  $\Delta(d, n) \leq n^{1 + \epsilon}$

  holds for $n \geq 2^{32d/\epsilon^2}$, for any fixed $\epsilon > 0$ (Gallagher et al., 2016, Mizuno et al., 2016).

The regime $n \leq O(d)$ ("head" regime) sees no improvement from tail analyses and remains governed by classical bounds and sporadic computer-verified cases.

3. Thresholds, Base Cases, and Handling of Small $n$

Tail bounds become effective only once $n$ crosses specific, dimension-dependent thresholds:

• For the tail-quasipolynomial bound on polyhedra, $n > d > 3$ suffices, but the sharper form requires $n \ge 2d$.
• For polytopes, the tail bound applies for $n > d$; exceptional cases with $d < 10$ or $n < d+10$ are verified computationally.
• For normal complexes, $n > d \ge 4$ is sufficient, with $n < d+8$ checked by finite enumeration.

For small $n$ (the "head"), induction reduces the problem to lower dimensions via $\Delta(d, n) \leq \Delta(d-1, n-1)$, ultimately recapitulating Hirsch-type linear bounds.

4. Proof Techniques and Inductive Structure

The tail bounds leverage recursive inequalities and combinatorial arguments:

• The Kalai–Kleitman recursion expresses $\Delta(d, n)$ via lower-dimensional and smaller-$n$ cases.
• Simultaneous induction on $d$ and $n$ utilizes the ansatz $g(d,n) = a^{d-3}(n-d)^k$, with parameters $a$, $k$ chosen to absorb higher-order terms:

$$\Delta(d, n) \leq g(d, n)\left[a^{-1} + 2^{1-k} + 2a^{3-d}(n-d)^{-k}\right]$$

Closing the induction requires $a^{-1} + 2^{1-k} \leq 1$ and $n$ large so $2a^{3-d}(n-d)^{-k}$ is negligible.

• For tail-cubic bounds, $a=2,~k=2$ yield $\Delta(d, n) \leq 2^{d-3}(n-d)^2$, which for $n-d \ge 2^{d-3}$ gives $\Delta(d, n) \leq (n-d)^3$.
• Tail-almost-linear bounds use $k = 1+\varepsilon/2$, $a=2^{\varepsilon/2}$, with $n-d \geq 2^{(d-3)/(\varepsilon/2)}$ sufficing for closure:

$\Delta(d, n) \leq (n-d)^{1+\varepsilon}$

For polyhedra, the combinatorial insight that any two vertices can be joined by a path passing through an intermediate facet is central to the recursive decomposition. For normal simplicial complexes, analogous dual-graph diameter arguments apply (Gallagher et al., 2016, Mizuno et al., 2016).

5. Diameter Bounds in Ricci Flow: Head and Tail

Analogous head-tail diameter bounds in geometric analysis provide upper and lower estimates for the diameter of compact Riemannian manifolds evolving under Ricci flow or for static metrics:

• Upper bound ("head"): There exists $C=C(n, A, B)$ such that for all $t \in [0, T)$,

$$\operatorname{diam}_t \leq C \left[1 + \mathrm{Vol}_t + \int_M R_{+}^{(n-1)/2}(x, t) \,d\mu_t\right]$$

where the constants $A,B$ arise from a Sobolev inequality or can be replaced by Yamabe invariants.

• Lower bound ("tail"): There exists $c>0$ such that for all $t \in [0,T)$,

$$\operatorname{diam}_t \geq c\ \exp(-a t - B' t \|R_{-}(\cdot, 0)\|_{\infty})[1 + t\|R_{-}(\cdot,0)\|_{\infty}]^{-1/2}\ [\mathrm{Vol}(M, g(0))]^{1/n}$$

where $a, B'$ depend on $(A, B)$ and the infimum of Perelman’s $F$-entropy; $B'=0$ if the initial scalar curvature is nonnegative. The constants $C$ and $c$ also depend only on the Sobolev or Yamabe constants and volume comparison factors (Zhang, 2013).

Proofs synthesize Sobolev/Poincaré-type inequalities, local geometric non-collapsing, and lower bounds for the conjugate-heat kernel along the flow.

6. Comparison to Previous Bounds and Sharpness

Prior to the tail bound paradigm, the best-known polyhedral diameter bounds exhibited worse asymptotics:

• Classical bound (Kalai–Kleitman 1992): $A_u(d, n) = O\left(n^{1+\log_2 d}\right)$ for polyhedra.
• Todd (2014): Improved this to $A_u(d, n) \leq (n-d)^{\log_2(d-1)}$.
• Sukegawa–Kitahara (2015): Delivered a tail-quasipolynomial bound valid for $n \ge 2d$.

Current tail bounds extend to polytopes and normal simplicial complexes, with explicit attention to first-principles verification in small regimes and the derivation of subcubic or near-linear tail-polynomial bounds for polyhedra (Gallagher et al., 2016).

In Ricci flow, the upper and lower diameter bounds are sharp in various regimes:

• For shrinking round spheres under Ricci flow, the lower "tail" bound precisely tracks the observed rate.
• On static hyperbolic quotients, the "head" bound captures the correct volume dependence (Zhang, 2013).

7. Practical Computability and Regime-Dependent Behavior

Tail diameter bounds for polyhedra and polytopes become effective only in the asymptotic regime where $n$ is large relative to $d$. For small $n$ (the "head"), no new improvements from tail techniques arise; computation relies on induction or finite enumeration. For Ricci flow, the constants in the head and tail bounds are, in principle, computable—especially when the Sobolev constants or the injectivity radius and curvature bounds are known. In Kähler–Ricci flow on Fano manifolds, diameter remains uniformly bounded due to uniform bounds on scalar curvature and volume.

A plausible implication is that, across both combinatorial and geometric settings, "tail" diameter bounds capture the growth rate in high-complexity regimes while "head" analysis must resort to direct or computational methods for low-complexity cases. This parallel structure underscores the universality of the head-tail framework in estimation of metric invariants across discrete and continuous geometric objects.