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arxiv: 2606.30241 · v1 · pith:Q6ED4XOFnew · submitted 2026-06-29 · 🧮 math.MG

On the Perelman-Pukhov quotient of successive radii: better and asymptotically optimal bounds

Bernardo Gonz\'alez Merino , Beatriz Mar\'in Gimeno , Mia Runge This is my paper

Pith reviewed 2026-06-30 03:20 UTC · model grok-4.3

classification 🧮 math.MG
keywords convex bodiessuccessive radiiPerelman-Pukhov quotientinner radiusouter radiusdiameterminimal width
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The pith

The Perelman-Pukhov quotient of successive radii is bounded strictly below i+1 for several (n,i) pairs, with the correct order in n when i equals n minus a fixed constant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper improves the classical upper bound of i+1 on the ratio of the (n-i+1)th successive outer radius to the ith successive inner radius of a convex body. It derives strictly smaller upper bounds for i=3 when 4 ≤ n ≤ 8, for i=4 when n=5 or 6, for i=5 when n=6, for i=6 when n=7, and for every i at least n minus order log n, by relating the radii to diameter and minimal width. A further sharpening holds for the single case n=5 and i=3 by incorporating an optimal three-dimensional inradius inequality. A sympathetic reader would care because these quotients govern the interplay between inner and outer approximations of convex bodies, which affects volume computations and geometric algorithms in varying dimensions.

Core claim

Perel'man and Pukhov proved the quotient is at most i+1. The paper shows the quotient is at most a strictly smaller value than i+1 in the cases i=3 for 4≤n≤8, i=4 for n=5 and 6, i=5 for n=6, i=6 for n=7, and for all i ≥ n - Θ(log n). The new bounds have the correct order of growth with n when i = n - m for any fixed m and n growing. The n=5 i=3 case receives an additional improvement by refining an earlier idea and using the optimal lower bound on the inradius in three dimensions in terms of circumradius and diameter.

What carries the argument

Relations that connect successive inner and outer radii to the diameter and minimal width of the convex body, with an extra application of the optimal three-dimensional inradius lower bound in the n=5 i=3 case.

If this is right

  • For i=3 the quotient is bounded by a number strictly less than 4 when the dimension is between 4 and 8.
  • Analogous strict improvements hold for the other listed small values of (n,i).
  • When i equals n minus a fixed m the upper bound grows linearly with n, which is the right order.
  • The n=5 i=3 case admits an even smaller explicit bound coming from three-dimensional geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same diameter-and-width technique could be checked on other radius sequences to see whether similar sharpenings appear.
  • Numerical sampling of convex bodies in dimensions 4 to 8 could reveal how close the new bounds are to being attained.
  • The fact that the order is optimal when i is n minus constant suggests the classical bound cannot be improved by more than a constant factor in that regime.

Load-bearing premise

The derivations rely on the auxiliary inequalities that link successive radii to diameter and minimal width being sufficiently tight, and on the three-dimensional inradius bound being optimal for the n=5 i=3 case; if those relations are weaker than assumed, the claimed improvements do not hold.

What would settle it

A convex body in dimension 4 for which the ratio of the second successive outer radius to the third successive inner radius meets or exceeds the paper's stated new upper bound would falsify the improvement in that case.

read the original abstract

Perel'man in 1987 and independently Pukhov in 1979 proved that the quotient between the $(n-i+1)$-th successive outer radius and the $i$-th successive inner radius of a convex body in $n$-dimensions is not larger than $i+1$. Apart from the solved cases by Jung 1901 $(i=1)$ and Steinhagen 1921 $(i=n)$, only Perel'man (1987, $n=3$, $i=2$) and Gonz\'alez Merino (2017, $n\geq 4$, $i=2$ and $i=n-1$) provided small improvements that beat this bound. In this paper, we obtain sharper inequalities using relations between these inner and outer measures with the diameter and minimal width. We improve the current bounds in the following cases: $i=3$ when $4\leq n \leq 8$, $i=4$ when $n=5$, $6$, $i=5$ when $n=6$, $i=6$ when $n=7$, and for every $i\geq n-\Theta(\log n)$. Notably, our bounds provide the right order in $n$ when $i=n-m$, with $m$ constant and $n$ arbitrarily large. Additionally, we improve the case $n=5$, $i=3$ even further by refining an idea of Perel'man and using the optimal lower bound of the inradius in terms of the circumradius and the diameter in 3-space (see [7]).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes improved upper bounds, strictly less than i+1, on the Perelman-Pukhov quotient between the (n-i+1)-th successive outer radius and the i-th successive inner radius of convex bodies in R^n. The improvements are obtained by relating the successive radii to the diameter and minimal width (with an additional refinement for the n=5, i=3 case that imports the optimal 3D inradius bound from reference [7]). The paper also derives bounds of the correct order in n when i = n-m for fixed m and n large.

Significance. If the derivations hold, the results advance convex geometry by supplying the first explicit improvements beyond the classical Jung, Steinhagen, Perelman, and González Merino cases in several dimensions, together with the first asymptotically sharp order for fixed codimension m. The approach relies on inequality chaining from established facts rather than parameter fitting, which strengthens the contribution when the auxiliary relations apply as stated.

minor comments (3)
  1. [Abstract and §1] The abstract and introduction would benefit from an explicit table or list of the new numerical bounds for each (n,i) pair rather than only describing the ranges of improvement.
  2. [§3 and §4] Notation for the successive radii R_k and r_k should be recalled or referenced at the start of each proof section to aid readability.
  3. [§2] The dependence on the minimal-width and diameter inequalities (cited from prior literature) could be stated more explicitly when they are first invoked, including the precise constants used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its significance in advancing bounds on Perelman-Pukhov quotients, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: bounds derived from external geometric inequalities

full rationale

The paper improves Perelman-Pukhov bounds by chaining relations between successive radii, diameter, and minimal width, plus one external 3D inradius result from reference [7]. No step reduces a claimed bound to a fitted parameter on the same data, a self-definition, or a load-bearing self-citation whose content is unverified. Prior self-citations (e.g., González Merino 2017) address only previously solved cases and are not invoked to justify the new inequalities. The derivation chain is therefore self-contained against independent geometric facts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are visible. The work rests on standard convex-geometry facts (Jung-Steinhagen, diameter-width relations) plus one imported 3D result.

pith-pipeline@v0.9.1-grok · 5838 in / 1131 out tokens · 32466 ms · 2026-06-30T03:20:47.080586+00:00 · methodology

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