On a Simplex Inscribed in a Ball

Mikhail Nevskii

arxiv: 2505.15739 · v2 · pith:UILEJCHYnew · submitted 2025-05-21 · 🧮 math.MG

On a Simplex Inscribed in a Ball

Mikhail Nevskii This is my paper

Pith reviewed 2026-05-22 13:50 UTC · model grok-4.3

classification 🧮 math.MG

keywords simplexinscribed simplexunit ballsuitable faceminimum-volume ellipsoidbarycenteropposite facen-dimensional geometry

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The pith

If a vertex of a simplex inscribed in the unit ball is suitable, then suitable faces of every dimension contain that vertex.

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

The paper studies nondegenerate simplices whose vertices all lie on the sphere of the unit ball in n dimensions. It recalls that every such simplex has at least one suitable face in each dimension m from 0 to n-1, where a face G is suitable when the intersection of the line through the centers of gravity of G and its opposite face with the boundary of the minimum-volume enclosing ellipsoid lies inside the unit ball. The new result shows that when one particular vertex meets the suitability condition, it is possible to choose a suitable face of each lower dimension that includes this vertex. This localizes the existence result around a given vertex rather than asserting only global existence.

Core claim

For a simplex S inscribed in the unit ball B_n, if a vertex of S is suitable, then for every m between 0 and n-1 there exists an m-dimensional face of S that contains the vertex and is suitable.

What carries the argument

The suitability condition for an m-dimensional face G, defined by whether the intersection point y of the line joining the barycenters of G and its opposite face with the boundary of the minimum-volume enclosing ellipsoid E lies inside the unit ball.

If this is right

For each dimension m from 0 to n-1 a suitable m-face containing the given suitable vertex can be selected.
The selection of suitable faces can be anchored at any suitable vertex rather than chosen independently for each dimension.
The property supplies a vertex-centered version of the earlier global existence statement.

Where Pith is reading between the lines

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

One could try to build a nested sequence of suitable faces all sharing the same suitable vertex.
The localization result may simplify inductive arguments that proceed by fixing a vertex and descending in dimension.
Checking the result on the regular simplex inscribed in the ball would give a concrete test case for small n.

Load-bearing premise

The simplex is nondegenerate with all vertices on the sphere, and the prior unconditional existence of suitable faces continues to hold.

What would settle it

An explicit inscribed simplex in dimension 3 that possesses a suitable vertex but no suitable edge or triangular face containing that vertex.

read the original abstract

Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum volume which contains $S$. Suppose $S\subset B_n$, $0\leq m\leq n-1$. Let $G$ be any $m$-dimensional face of $S$ and let $H$ be the opposite $(n-m-1)$-dimensional face. Denote by $g$ and $h$ the centers of gravity of $G$ and $H$ respectively. Define $y$ as the intersection point of the line passing from $g$ to $h$ with the boundary of $E$. Let us call the face $G$ suitable if $y\in B_n.$ Earlier it was proved that each simplex $S\subset B_n$ has a suitable face of any dimension $\leq n-1$. We show the following. Let $S$ be inscribed in $B_n$. If some vertex of $S$ is suitable, then there exists a suitable face of any dimension $\leq n-1$ which contains this vertex.

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.

Desk Editor's Note private letter to a colleague

This extends the prior existence result for suitable faces by showing that suitability at one vertex implies suitable containing faces in every lower dimension.

read the letter

The main point is a conditional strengthening of the earlier theorem on suitable faces of a simplex inscribed in the unit ball. If a vertex satisfies the suitability condition, then suitable faces exist in every dimension up to n-1 that contain it. The argument adapts the same minimum-volume ellipsoid E and the line through centers of gravity of opposite faces, checking where it meets the boundary of E and whether that point stays inside the ball. By restricting to faces through the fixed suitable vertex, the construction produces the desired faces without new parameters or adjustments. The definitions line up with the prior unconditional result, and the steps follow directly from it. No degeneracy or positioning issues for E appear in the given construction. A minor limitation is the dependence on the base existence theorem holding for the whole simplex. If that earlier result has any boundary cases, they would carry through here, though the paper does not claim to resolve them. The citation pattern is appropriate, pointing back to the unconditional version without circularity. This is aimed at readers already working in convex or metric geometry who track refinements of simplex properties. Someone familiar with the background will see the value in the localization right away. It is a modest but precise addition rather than a broad new direction. I would bring it to a reading group focused on geometric inequalities for discussion. It deserves peer review because the central claim is grounded in the definitions and follows logically from the supplied prior result.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a refinement of an earlier existence result for 'suitable' faces of a nondegenerate simplex S inscribed in the unit ball B_n. With E the minimum-volume ellipsoid containing S, an m-face G is suitable if the intersection y of the line through the barycenters g of G and h of the opposite (n-m-1)-face with the boundary of E lies inside B_n. The main result states that if a given vertex v of S is suitable, then for every dimension m from 0 to n-1 there exists a suitable m-face containing v.

Significance. If correct, the result strengthens the prior unconditional existence theorem by localizing the choice of suitable faces to those containing a prescribed suitable vertex. This structural property may be useful in convex geometry for constructions involving the John or Löwner ellipsoid of a simplex and for understanding how suitability propagates across face dimensions. The argument relies on the definitions of suitability and the prior theorem without introducing free parameters or post-hoc adjustments.

minor comments (3)

The reference to the earlier unconditional existence result should be cited explicitly (e.g., as a numbered reference in the bibliography) rather than stated only as 'earlier it was proved.'
Add a brief remark or diagram illustrating the line through g and h and the point y for the case m=0 (the vertex itself) to clarify the base case of the induction or selection argument.
In the statement of the main theorem, explicitly note that the opposite face to an m-face has dimension n-m-1 to avoid any ambiguity in the dimension count.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the main result, and the recommendation for minor revision. We are pleased that the referee recognizes the refinement of the earlier existence theorem for suitable faces.

Circularity Check

0 steps flagged

Minor self-citation of prior existence result; central claim remains independent

full rationale

The paper defines suitability of an m-face G via the minimum-volume ellipsoid E containing S, the centers of gravity g and h of G and its opposite face H, and the intersection y of the line gh with the boundary of E, calling G suitable precisely when y lies inside the unit ball B_n. It invokes an earlier unconditional result that every nondegenerate simplex inscribed in B_n possesses at least one suitable face of each dimension m ≤ n-1. The new theorem then asserts that, when S is inscribed, suitability of a single vertex v implies the existence of suitable faces of every dimension that contain v. The argument proceeds by using the given suitability of v to select, for each m, an m-face containing v whose opposite-face line segment satisfies the E-boundary condition inside B_n. No equation equates the new claim to its own inputs by construction, no parameter is fitted and then renamed as a prediction, and the prior existence statement supplies external support rather than a load-bearing self-referential loop. The derivation therefore adds independent geometric content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the definition of the minimum-volume enclosing ellipsoid, centers of gravity of faces, the line through g and h, and the prior unconditional existence theorem for suitable faces.

axioms (2)

domain assumption The minimum-volume ellipsoid containing a simplex is well-defined and unique for nondegenerate simplices.
Invoked in the definition of y and the suitability condition.
standard math Centers of gravity of faces exist and the line through them intersects the ellipsoid boundary.
Standard convex geometry facts used to define y.

pith-pipeline@v0.9.0 · 5748 in / 1206 out tokens · 36413 ms · 2026-05-22T13:50:00.539619+00:00 · methodology

On a Simplex Inscribed in a Ball

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)