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arxiv: 2604.26228 · v1 · submitted 2026-04-29 · 🧮 math.OC

On the geometry of circumcentric directions of cones

Yunier Bello-Cruz This is my paper

Pith reviewed 2026-05-07 13:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords normconeballalongconesinscribedadmissiblecircumcentric
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The pith

The circumcentric direction of a cone has an exact polyhedral set of admissible perturbations that is larger than its inscribed ball in the polar cone.

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

The paper refines the understanding of the circumcentric direction d for a finitely generated polyhedral cone. It establishes that the set of vectors that can perturb d while preserving key geometric properties forms a polyhedron. This polyhedron strictly contains the ball of radius equal to the squared norm of d inside the polar cone and extends unboundedly in the direction of the polar cone. Using this structure, the paper derives a closed-form expression for the squared norm of d based on the inverse of the Gram matrix of the cone's generating vectors, along with matching spectral bounds and an identity showing that the norm equals the cosine of the angle between the generators and the negative normalized direction. The inscribed ball property is shown to hold for general closed convex pointed cones when the affine hull of the normalized extremal section avoids the origin.

Core claim

The exact set of admissible perturbations for the circumcentric direction d of a polyhedral cone K is a polyhedron contained in the polar cone K^polar. This set is strictly larger than the inscribed ball of radius ||d||^2 and is unbounded along K^polar. It provides a closed-form formula for ||d||^2 expressed using the inverse Gram matrix of the conic base vectors, together with two-sided bounds from the spectrum of that matrix. An aperture identity holds: ||d|| equals the cosine of the angle theta formed by the generators with the axis given by -d normalized. For general closed convex pointed cones, the same inscribed-ball containment holds if the normalized extremal section E_K has affine h

What carries the argument

The polyhedral admissible perturbation set for the circumcentric direction, which refines the inscribed ball of radius ||d||^2 in the polar cone and enables the closed-form norm expression.

Where Pith is reading between the lines

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

  • The explicit formulas allow direct computation of the circumcentric direction length from the generators without solving an auxiliary optimization problem.
  • The parallel-resistance rule for direct-product cones implies that the overall norm can be obtained by combining individual component norms in a simple reciprocal sum.
  • The clean obstruction for p-cones with p not equal to 2 shows that the inscribed-ball property requires the stated affine-hull condition and does not hold universally.

Load-bearing premise

The normalized extremal section of the cone has an affine hull that does not contain the origin.

What would settle it

Construct a closed convex pointed cone whose normalized extremal section has affine hull containing the origin and check whether a ball of radius equal to the squared circumcentric norm still fits inside the polar cone without intersecting the cone itself.

Figures

Figures reproduced from arXiv: 2604.26228 by Yunier Bello-Cruz.

Figure 1
Figure 1. Figure 1: The orthant case of Theorem 3.3. The exact admissible set P = {v1 ≤ 1 2 , v2 ≤ 1 2 } from Proposition 3.1 is the unbounded L-shaped region (blue); the inscribed-ball estimate B = {∥v∥ ≤ 1 2 } of [13] is the disc (red), tangent to ∂P only at 1 2 e1 and 1 2 e2. Along the diagonal w = (e1 +e2)/ √ 2, the inscribed ball reaches 1 2w (short red arrow) while Corollary 3.2 delivers ρK(w) = 1/ √ 2, taking 0 to the … view at source ↗
Figure 2
Figure 2. Figure 2: The second-order cone case of Theorem 6.7 for n = 3. The upper (blue) cone is K = L 3 ; its normalized extreme-ray set EK is the unit circle on the cone boundary at height t = 1/ √ 2. The horizontal plane through that circle is cl aff(EK); it lies at positive distance from the origin, so the hypothesis of Theorem 6.3 is satisfied. The polar cone K◦ = −L3 (red, opening downward) contains the projection-base… view at source ↗
Figure 3
Figure 3. Figure 3: The non-example of Theorem 6.14. The first three normalized extreme rays e1, e2, e3 lie on the simplex plane ∆ = {x1 + x2 + x3 = 1} (shaded), but u 4 has component sum 1/ √ 3 and therefore lies on the strictly parallel plane {x1 + x2 + x3 = 1/ √ 3}, off ∆. The four points are then affinely independent in R3 , their convex hull is a non-degenerate tetrahedron (edges to u 4 dashed), and aff(EK) = R3 ∋ 0. The… view at source ↗
read the original abstract

Behling, Bello-Cruz, Lara-Urdaneta, Oviedo, and Santos showed that the circumcentric direction $d$ of a finitely generated polyhedral cone $\KK\subset\RR^n$ admits an inscribed Euclidean ball of radius $\norm{d}^2$ inside the polar cone $\Kpolar$. We sharpen this result in several ways. The exact set of admissible perturbations is a polyhedron, strictly larger than the inscribed ball off the generators and unbounded along $\Kpolar$. From it we read off a closed form for $\norm{d}^2$ in terms of the inverse Gram matrix of the conic base, with two-sided spectral bounds, and an aperture identity $\norm{d}=\cos\theta$ relating the generators to the axis $-d/\norm{d}$. The inscribed-ball estimate extends to closed convex pointed cones under one geometric condition: the normalized extremal section $E_\KK$ has affine hull avoiding the origin. The admissible set is then the intersection of half-spaces indexed by $E_\KK$, and the inscribed ball touches its boundary along $\norm{d}^2\,\closu E_\KK$. A Jordan-frame argument verifies the hypothesis for every simple symmetric cone and gives $\norm{d}^2=1/r$ for the Jordan rank $r$; the same value $1/n$ shows up for the doubly nonnegative cone, the direct-product case obeys the parallel-resistance rule $1/\norm{d}^2=\sum_\ell 1/\norm{d_\ell}^2$, and the $p$-cones with $p\ne 2$ provide a clean obstruction. We close with a sharp formula for the largest step from $d$ along a prescribed direction, worked out for $L_\infty$-ball constrained least squares and second-order cone programming; a piecewise smooth version where the inner Slater condition is exactly Mangasarian--Fromovitz; and a Bregman analogue covering a Mahalanobis instance and a mirror-descent step.

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

2 major / 2 minor

Summary. The manuscript sharpens the inscribed-ball result of Behling et al. on the circumcentric direction d of a finitely generated polyhedral cone K by showing that the exact admissible perturbation set is a polyhedron strictly larger than the ball and unbounded along the polar; it derives a closed-form expression for ||d||^2 from the inverse Gram matrix of the conic base together with two-sided spectral bounds and the aperture identity ||d|| = cos θ. The inscribed-ball estimate is extended to general closed convex pointed cones provided the affine hull of the normalized extremal section E_K avoids the origin, with the admissible set then the intersection of half-spaces indexed by E_K. Specific evaluations are given for simple symmetric cones (||d||^2 = 1/r), the doubly nonnegative cone (1/n), direct products (parallel-resistance rule), and an obstruction for p-cones with p ≠ 2; the paper closes with sharp largest-step formulas from d, including applications to L_∞-constrained least squares and second-order cone programming.

Significance. If the derivations hold, the work supplies sharper geometric and algebraic characterizations of circumcentric directions than the prior inscribed-ball estimate, together with explicit, computable formulas and cone-specific identities that are directly relevant to algorithm analysis in conic optimization. The Jordan-frame verification for symmetric cones, the parallel-resistance rule, and the concrete step-size formulas for LS and SOCP constitute concrete strengths that could aid both theory and implementation.

major comments (2)
  1. [Derivation of the closed form for ||d||^2] The closed-form expression for ||d||^2 is read off from the inverse Gram matrix of the conic base. For a general finitely generated polyhedral cone the extreme-ray generators may be linearly dependent or exceed dimension n, rendering the Gram matrix singular. The manuscript provides no indication whether the base is restricted to a linearly independent spanning set, whether a pseudo-inverse is intended, or how the formula and spectral bounds are modified under rank deficiency. This directly affects the claimed closed form and its generality.
  2. [Extension to closed convex pointed cones] The extension of the inscribed-ball estimate to arbitrary closed convex pointed cones rests on the hypothesis that the affine hull of the normalized extremal section E_K avoids the origin. While the Jordan-frame argument verifies the hypothesis for simple symmetric cones, the manuscript does not clarify whether the condition is necessary in general, whether it holds for all pointed cones, or supply counter-examples when it fails; the polyhedron description of the admissible set and the touching property along ||d||^2 closu E_K therefore rest on an unexamined assumption for the broader class.
minor comments (2)
  1. The abstract lists applications to L_∞-ball constrained least squares and SOCP; the corresponding sections should explicitly derive and verify the piecewise-smooth largest-step formula and confirm that the inner Slater condition coincides with the Mangasarian–Fromovitz constraint qualification.
  2. Notation for the polar cone, the circumcentric direction d, and the normalized extremal section E_K should be introduced with a single consistent definition in the introduction rather than piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below and plan to incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: The closed-form expression for ||d||^2 is read off from the inverse Gram matrix of the conic base. For a general finitely generated polyhedral cone the extreme-ray generators may be linearly dependent or exceed dimension n, rendering the Gram matrix singular. The manuscript provides no indication whether the base is restricted to a linearly independent spanning set, whether a pseudo-inverse is intended, or how the formula and spectral bounds are modified under rank deficiency. This directly affects the claimed closed form and its generality.

    Authors: We appreciate this observation. The derivation in the manuscript implicitly assumes that the conic base consists of a linearly independent set of generators for the cone. This is standard and without loss of generality, as one can always extract a linearly independent spanning subset from the extreme rays (a basis for the linear span of the cone). The Gram matrix is then square and invertible. We will revise the manuscript to explicitly state this assumption in the section introducing the closed-form expression and the spectral bounds. Under rank deficiency, the formula applies to any choice of basis, yielding the same ||d||^2 since the circumcentric direction is unique. No pseudo-inverse is required, and the two-sided spectral bounds remain valid as they derive from the eigenvalues of the inverse Gram matrix of the basis. revision: yes

  2. Referee: The extension of the inscribed-ball estimate to arbitrary closed convex pointed cones rests on the hypothesis that the affine hull of the normalized extremal section E_K avoids the origin. While the Jordan-frame argument verifies the hypothesis for simple symmetric cones, the manuscript does not clarify whether the condition is necessary in general, whether it holds for all pointed cones, or supply counter-examples when it fails; the polyhedron description of the admissible set and the touching property along ||d||^2 closu E_K therefore rest on an unexamined assumption for the broader class.

    Authors: We thank the referee for highlighting this. The geometric condition is indeed an assumption required for the extension to general closed convex pointed cones, ensuring that the admissible perturbation set is a polyhedron whose boundary is touched by the inscribed ball along the closure of E_K. It is not satisfied by every pointed cone; for example, it fails for certain cones where the extremal rays or section are positioned such that their normalized points have an affine dependence including the origin. However, as shown in the paper, it holds for all simple symmetric cones via the Jordan frame decomposition. We will revise the relevant section to clarify the necessity of the condition for the polyhedral description and note that it does not hold universally for all pointed cones, although we focus on the cases (such as symmetric cones) where it is verified. This strengthens the statement without altering the main results. revision: partial

Circularity Check

0 steps flagged

Derivations rely on standard convex geometry and linear algebra without reduction to inputs

full rationale

The paper cites prior work (including co-authors) only for the initial inscribed-ball property of radius ||d||^2 inside the polar cone, then independently sharpens this by characterizing the exact admissible perturbation set as a polyhedron strictly larger than the ball. From this polyhedral description it algebraically derives the closed-form ||d||^2 expression via the inverse Gram matrix of the generators, two-sided spectral bounds, and the aperture identity ||d|| = cos theta. These steps use direct convex-analytic and linear-algebraic manipulation on the cone generators and their inner products; they do not reduce by construction to the cited ball radius, to any fitted parameter, or to a self-referential definition. Extensions to symmetric cones, p-cones, and optimization instances follow from the same geometric construction without invoking uniqueness theorems or ansatzes from the authors' prior work. The Gram-matrix step is a standard algebraic consequence once the polyhedron is obtained and assumes the base permits inversion (a modeling choice, not a circularity). No load-bearing self-citation or self-definitional pattern appears in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of Euclidean convex geometry and polar cones; no free parameters or invented entities are introduced beyond the geometric condition on the extremal section.

axioms (2)
  • standard math Standard properties of finitely generated polyhedral cones, their polars, and inscribed balls in Euclidean space
    Invoked throughout for definitions of circumcentric direction, admissible perturbations, and the inscribed-ball estimate.
  • domain assumption Affine hull of normalized extremal section E_K avoids the origin
    Explicitly stated as the geometric condition needed to extend the inscribed-ball result to general closed convex pointed cones.

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Reference graph

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