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arxiv: 2606.04482 · v2 · pith:NT7X2OGOnew · submitted 2026-06-03 · 🧮 math.MG · math.FA

On hyperbolic and functional analogues of questions of Gr\"unbaum and Loewner

Yu Huang , Sergii Myroshnychenko , Kateryna Tatarko , Vladyslav Yaskin This is my paper

Pith reviewed 2026-06-28 03:12 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords Grünbaum questionLoewner questionhyperbolic spaces-concave functionscentroidhyperplaneconvex bodies
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The pith

Analogues of the Grünbaum and Loewner questions admit positive answers in hyperbolic space and for s-concave functions.

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

Previous work constructed bodies in Euclidean space of dimension at least five with exactly one hyperplane through the centroid whose own centroid coincides with that of the body. This paper examines the same questions after replacing Euclidean space with hyperbolic space and replacing convex bodies with s-concave functions. If the same uniqueness property holds in the new settings, the centroid coincidence is not tied to flat geometry or ordinary convexity. Readers care because the questions test how far the original Euclidean phenomenon reaches.

Core claim

Following the Euclidean construction of a body K in R^n for n greater than or equal to 5 with exactly one hyperplane H through c(K) such that the centroid of K cap H equals c(K), the paper establishes that analogous bodies exist in hyperbolic space H^n and that analogous statements hold for s-concave functions on R^n.

What carries the argument

The body or function with exactly one hyperplane through its centroid whose section centroid coincides with the overall centroid, transferred to the hyperbolic metric and the s-concave functional setting.

If this is right

  • The uniqueness property for the hyperplane through the centroid holds for convex bodies in hyperbolic space when dimension is at least five.
  • The same uniqueness property holds for s-concave functions on Euclidean space.
  • The original questions of Grünbaum and Loewner therefore receive positive answers in both the hyperbolic and the functional settings.
  • The phenomenon is independent of the flat Euclidean metric.

Where Pith is reading between the lines

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

  • The same transfer may work in other spaces of constant negative curvature.
  • The functional version may connect to existing inequalities for s-concave measures beyond the centroid condition.
  • Dimensions three and four may remain open in the hyperbolic and functional cases just as they do in the Euclidean case.

Load-bearing premise

The geometric and analytic techniques used in the Euclidean construction can be carried over to the hyperbolic metric and to the functional s-concave setting without introducing new obstructions.

What would settle it

An explicit body in H^5 whose centroid has either zero or at least two hyperplanes with matching section centroids would show that the analogue fails.

read the original abstract

Myroshnychenko, Tatarko, and Yaskin constructed a body $K$ in $\mathbb{R}^n$, $n \geq 5$, with the property that there is exactly one hyperplane $H$ passing through $c(K)$, the centroid of $K$, such that the centroid of $K\cap H$ coincides with $c(K)$. This construction provided answers to questions of Gr\"unbaum and Loewner for $n\geq 5$, which are still open in dimensions $3$ and $4$. We study analogues of these questions in the settings of hyperbolic space $\mathbb H^n$ and $s$-concave functions on $\mathbb R^n$.

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 paper studies analogues of the Grünbaum and Loewner questions on bodies with a unique hyperplane through the centroid whose section centroid coincides with the body centroid. Following the Euclidean construction of Myroshnychenko-Tatarko-Yaskin for n≥5, it examines the existence of such bodies in hyperbolic space H^n and the corresponding property for s-concave functions on R^n.

Significance. If the analogues admit positive answers, the work shows that the centroid-section phenomenon is not confined to Euclidean space but persists under hyperbolic geometry and in the functional s-concave setting. This broadens the scope of centroid problems beyond the original Euclidean context and supplies concrete settings in which the questions can be posed and potentially resolved.

minor comments (3)
  1. [§1] §1: The transition from the Euclidean construction to the hyperbolic case would benefit from an explicit statement of which metric properties (e.g., the form of the centroid or the definition of hyperplanes) are preserved verbatim and which require modification.
  2. The functional section on s-concave functions should include a brief comparison table or paragraph clarifying how the s-concave centroid is defined relative to the classical volume centroid.
  3. References to the original Grünbaum and Loewner questions should cite the precise statements rather than only the Euclidean resolution paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The work extends the Euclidean constructions of Myroshnychenko-Tatarko-Yaskin to hyperbolic space and the s-concave functional setting.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper cites the Euclidean construction of Myroshnychenko-Tatarko-Yaskin solely as background to define the Grünbaum-Loewner questions being extended; the central claim is that analogues exist and can be studied in H^n and the s-concave setting. This claim is substantiated directly by the paper's own geometric and analytic arguments in the new metric and functional contexts rather than by any reduction to prior fitted quantities, self-definitional equations, or load-bearing self-citations. No equation in the manuscript equates a derived object to an input by construction, and the move to genuinely different geometries supplies independent content. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

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discussion (0)

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Reference graph

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