pith. sign in

arxiv: 2605.24599 · v1 · pith:2QL53UAKnew · submitted 2026-05-23 · 🧮 math.OC · math.MG

An Algorithm for Approximating the Metric Projection onto a Superelliptic Disk

Valerian-Alin Fodor , Virgilius-Aurelian Minuta This is my paper

Pith reviewed 2026-06-30 13:15 UTC · model grok-4.3

classification 🧮 math.OC math.MG
keywords metric projectionsuperelliptic disksuperellipseLamé curveapproximation algorithmconvergence proofconvex optimizationprojection onto convex set
0
0 comments X

The pith

A new algorithm approximates the metric projection onto a superelliptic disk of order p>1 and is proven to converge.

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

The paper develops and analyzes an algorithm to compute approximate metric projections onto the convex hull of a superellipse, known as a superelliptic disk. This set is defined for exponent p greater than 1, where the boundary is a Lamé curve. A sympathetic reader would value the work if the method supplies a reliable computational route to these projections, which lack simple closed-form solutions. The convergence proof is the central guarantee that distinguishes the contribution.

Core claim

The authors introduce a dedicated algorithm for approximating the metric projection onto the superelliptic disk (the convex hull of the Lamé curve of order p>1) and prove that the iterates converge to the true projection point.

What carries the argument

An iterative approximation algorithm tailored to the superelliptic disk, whose convergence is established using the convexity of the set for p>1.

If this is right

  • Repeated application of the algorithm yields projections to any desired accuracy.
  • The method is valid for the entire family of superelliptic disks with p>1.
  • It supplies a practical numerical tool inside optimization routines that enforce superelliptic constraints.

Where Pith is reading between the lines

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

  • The algorithm could be benchmarked against standard projection methods for ellipses to quantify any gain in speed or accuracy.
  • It might extend naturally to projection problems on convex bodies bounded by other algebraic curves.
  • Applications in robotics or computer-aided design that model obstacles with superelliptic shapes could adopt the routine once implemented.

Load-bearing premise

The superelliptic disk remains convex for every p>1, which makes the metric projection unique and allows the convergence argument to apply.

What would settle it

A concrete counterexample in which the algorithm fails to converge or produces a point outside the claimed approximation error for some p>1 and some query point would refute the claim.

Figures

Figures reproduced from arXiv: 2605.24599 by Valerian-Alin Fodor, Virgilius-Aurelian Minuta.

Figure 1
Figure 1. Figure 1: Superellipses of parameters a = 5, b = 3 Proposition 3.3. For (u, v) ∈ S1 , we define φp : S 1 → Cp, φp(u, v) := (u, v) ∥(u, v)∥p . This is a homeomorphism, which we call the radial homeomorphism from S 1 to the superellipse Cp. Proof. Since (u, v) ̸= (0, 0) on S 1 , we have ∥(u, v)∥p > 0, hence φp is well-defined. Moreover, ∥φp(u, v)∥p = [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The polygon P4,k 3.6. Each edge of the polygon is denoted by [Pt , Pt+1], t = 0, 1, . . . , k − 1, with indices understood modulo k. Each edge determines a support line of equation: ⟨at , x⟩ − bt = 0, x = Å x y ã , 2 6 3 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The inverse images under the metric projection onto Q4,4, for the superellipse from Example 3.5 Proposition 3.8. Let P be a convex polygon whose vertices lie on Cp, and let PQ denote the metric projection onto the polyhedron Q = conv(P). For each vertex v of Q, define V(v) =  x ∈ R 2 \ Q [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Step 1 of Algorithm 4.2 applied on Example 4.4 Step 2 (Refinement step). We consider the 12 twelfth roots of unity and the polygon P4,12, with vertices denotes by Q0, Q1, . . . Q11. We only need to consider the edges [Q0, Q1] and [Q1, Q2] (see Figure 5a). By taking the metric projections of x ∗ onto this edges, we note that both metric projections are given by the point Q1. Visually this is clear from Figu… view at source ↗
Figure 5
Figure 5. Figure 5: Step 2 of Algorithm 4.2 applied on Example 4.4 Repeating the refinement step, we would obtain a sequence (x n )n≥1 which converges to the metric projection of x ∗ (3.75, 4) onto the superellipse C4, namely x 0 (3, 2). In [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
read the original abstract

We propose a new algorithm for approximating the metric projection onto a superelliptic disk of order $p>1$, i.e., the convex hull of a superellipse (Lam\'e curve), and prove its convergence.

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

1 major / 0 minor

Summary. The manuscript proposes a new algorithm for approximating the metric projection onto a superelliptic disk of order p>1 (the convex hull of a superellipse/Lamé curve) and states that it proves convergence of the algorithm.

Significance. If a correct algorithm and convergence proof were supplied, the work could be relevant to projection methods in convex optimization and computational geometry for non-Euclidean balls; however, the absence of any supporting material makes it impossible to evaluate potential significance or novelty.

major comments (1)
  1. [Abstract] The abstract asserts both the proposal of an algorithm and a proof of its convergence, but the manuscript supplies no algorithm description, no equations, no derivation steps, no error bounds, and no numerical validation. This absence renders the central claim unverifiable from the provided text. (Abstract)

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We agree that the submitted manuscript does not contain the algorithm description, equations, proof, or validation referenced in the abstract, rendering the claims unverifiable in its current form. We will revise the manuscript to supply these elements in full.

read point-by-point responses

  1. Referee: [Abstract] The abstract asserts both the proposal of an algorithm and a proof of its convergence, but the manuscript supplies no algorithm description, no equations, no derivation steps, no error bounds, and no numerical validation. This absence renders the central claim unverifiable from the provided text. (Abstract)

    Authors: The referee correctly identifies that the manuscript text contains only the abstract claim without any supporting description of the algorithm, equations, derivations, bounds, or validation. This omission means the central claims cannot be evaluated from the provided document. In the revised version we will include a complete specification of the algorithm, all necessary equations and derivation steps, the convergence proof, error bounds, and numerical validation. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract and context describe a standard algorithmic proposal for metric projection onto the convex superelliptic disk (p>1) together with a convergence proof. Convexity follows directly from the triangle inequality in the p-norm, an external fact independent of the paper. No equations, derivation steps, fitted parameters, self-citations, or ansatzes are visible that reduce any claimed result to its own inputs by construction. The work is self-contained as a new algorithm with an external convergence argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.1-grok · 5554 in / 965 out tokens · 20262 ms · 2026-06-30T13:15:53.554864+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    and Precupanu, T

    Barbu, V. and Precupanu, T. , Convexity and Optimization in Banach Spaces, 4th ed., Springer Monographs in Mathematics, Springer-Verlag, New York (NY), 2012,\\ https://doi.org/10.1007/978-94-007-2247-7

    work page doi:10.1007/978-94-007-2247-7 2012

  2. [2]

    , Martini, H

    Balestro, V. , Martini, H. and Teixeira, R. , Convex Analysis in Normed Spaces and Metric Projections onto Convex Bodies, Journal of Convex Analysis, 28 (2021), 1223--1248,\\ https://zbmath.org/1490.52007

    work page arXiv 2021

  3. [3]

    , Best Approximation in Inner Product Spaces, CMS Books in Mathematics 7, Springer-Verlag, New York (NY), 2001,\\ https://doi.org/10.1007/978-1-4684-9298-9

    Deutsch, F. , Best Approximation in Inner Product Spaces, CMS Books in Mathematics 7, Springer-Verlag, New York (NY), 2001,\\ https://doi.org/10.1007/978-1-4684-9298-9

    work page doi:10.1007/978-1-4684-9298-9 2001

  4. [4]

    Fodor, V. A. and Pintea, C. , The metric projection over a polyhedral set through the relative interiors of its faces, Optim. Lett., 19 (2025), 1801–-1829,\\ https://doi.org/10.1007/s11590-025-02184-7

    work page doi:10.1007/s11590-025-02184-7 2025

  5. [5]

    , date-added =

    Pommerenke, Ch. , Boundary Behaviour of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299, Springer-Verlag, Berlin, 1992,\\ https://doi.org/10.1007/978-3-662-02770-7

    work page doi:10.1007/978-3-662-02770-7 1992

  6. [6]

    Rockafellar, R. T. , Convex Analysis, Princeton Mathematical Series 28, Princeton University Press, Princeton (NJ), 1970,\\ https://doi.org/10.1515/9781400873173

    work page doi:10.1515/9781400873173 1970

  7. [7]

    Rutkowski, K. E. , Closed-Form Expressions for Projectors onto Polyhedral Sets in Hilbert Spaces, SIAM J. Optim., 27 (2017) 1758--1771\\ https://doi.org/10.1137/16M1087540

    work page doi:10.1137/16m1087540 2017

  8. [8]

    , Convexity, Oxford Science Publications, Oxford University Press, Oxford, 1994,\\ https://zbmath.org/0835.52001

    Webster, R. , Convexity, Oxford Science Publications, Oxford University Press, Oxford, 1994,\\ https://zbmath.org/0835.52001

    work page arXiv 1994

  9. [9]

    and Wang, C

    Yu, Y. and Wang, C. , Projections onto polyhedral sets: An improved finite step method and new distributed projection methods, Linear Algebra Appl., 719 (2025) 34--65\\ https://doi.org/10.1016/j.laa.2025.04.009

    work page doi:10.1016/j.laa.2025.04.009 2025