pith. sign in

arxiv: 2604.19479 · v1 · submitted 2026-04-21 · 🧮 math.AG · math.MG

Algebraic Distance Optimization in Polyhedral Norms

Eliana Duarte , Nidhi Kaihnsa , Julia Lindberg , Ang\'elica Torres , Madeleine Weinstein This is my paper

Pith reviewed 2026-05-10 01:58 UTC · model grok-4.3

classification 🧮 math.AG math.MG
keywords real algebraic varietiespolyhedral normsVoronoi conesmedial axisstratificationsemialgebraic setsdistance minimizationcodimension one
0
0 comments X

The pith

For codimension-one real algebraic varieties, distance minimization under a polyhedral norm reduces to a stratification by Voronoi cone dimensions that algebraically describes the medial axis.

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

The paper studies the problem of finding the shortest distance from a point to a real algebraic variety X inside Euclidean space when the distance comes from a polyhedral norm. For varieties of codimension one it partitions X into subsets of points that share the same Voronoi cone dimension, where that dimension equals the expected dimension of the Voronoi cell itself. The partition is shown to be a stratification whose pieces are all semialgebraic sets. The authors then derive an algebraic description of the medial axis, the locus of points that achieve their minimal distance at more than one place on X.

Core claim

For a real algebraic variety X of codimension one we decompose X into sets of points whose Voronoi cones have identical dimension, equal to the expected dimension of the Voronoi cell. This decomposition forms a stratification of X and each stratum is semialgebraic. The medial axis therefore admits an algebraic description.

What carries the argument

The Voronoi cone attached to each point of X, whose dimension is fixed by the normal space at the point together with the inner normal fan of the polyhedral ball.

If this is right

  • The decomposition of X is a stratification.
  • Every stratum is a semialgebraic set.
  • The medial axis of X admits an algebraic description.
  • Distance minimization to X reduces to algebraic computations within each stratum.

Where Pith is reading between the lines

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

  • The same cone-dimension grouping may supply a practical way to compute minimal distances by solving systems on each stratum separately.
  • The stratification could be used to locate singularities or special points of the medial axis without numerical sampling.

Load-bearing premise

The geometry of the Voronoi cell at each point on X is completely determined by the normal space at that point and the inner normal fan of the polyhedral ball.

What would settle it

A concrete codimension-one variety, for example a plane curve defined by a polynomial equation, on which the collection of constant-dimension Voronoi-cone sets fails to be a stratification or contains a stratum that is not semialgebraic.

Figures

Figures reproduced from arXiv: 2604.19479 by Ang\'elica Torres, Eliana Duarte, Julia Lindberg, Madeleine Weinstein, Nidhi Kaihnsa.

Figure 1
Figure 1. Figure 1: From left to right, we have P and its inner normal fan, and P ∗ and its inner normal fan. Example 2.7. Consider the three dimensional cube P = Conv{(1, 1, 1),(1, −1, 1),(−1, −1, 1),(−1, 1, 1),(1, 1, −1),(1, −1, −1),(−1, −1, −1),(−1, 1, −1)}. The inner normal fan consists of 8 three-dimensional cones (one for each vertex), which coincide with the open orthants; 12 two-dimensional cones (one for each edge), … view at source ↗
Figure 2
Figure 2. Figure 2: From left to right, we have P, the inner normal fan of P, P ∗ , and the inner normal fan of P ∗ . 2.2 Polyhedral Norms Let L be a finite set of linear functionals on R n such that the set P = {x ∈ R n | ℓ(x) ≤ 1 for ℓ ∈ L} is bounded and describes a full-dimensional convex polytope. Furthermore, we impose that L = −L so the polytope P is centrally symmetric around the origin, i.e. P = −P. For the entirety … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of Type(v) in Theorem 3.9. Proof. Let u ∈ R n \ X be a point in the Voronoi cell of v ∈ X. We claim u − v ∈ C(B∗, F∗) for some F ∈ Type(v). Let us assume this is not true so that u − v ∈ C(B∗ , T ∗ ) for some face T such that T ̸⊆ F for all F ∈ Type(v). Since u − v is minimized over the face T, we have that Bλ(u) intersects X at the point v which lies in the face T where λ := d(u, X). Hence, t… view at source ↗
Figure 4
Figure 4. Figure 4: Twisted cubic with the normal space depicted at [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Figures from Theorem 3.10. The polyhedral ball B is the cube and the variety X is the twisted cubic. The top row illustrates the type of v1 = (0, 0, 0) and the bottom row illustrates the type of v2 = (0.6, 0.6 2 , 0.6 3 ). Type(v1) is given by the blue faces of B in (a) and Type(v1) is given by the green faces of B in (e). The corresponding dual faces to those in Type(v1) and Type(v2) are depicted in blue … view at source ↗
Figure 6
Figure 6. Figure 6: The hyperboloid defined as V(36x 2 + 9y 2 − 4z 2 − 36). The point v = (1, 0, 0) is a saddle point. Its Voronoi cone is the x-axis depicted in black, but its Voronoi cell consists of the single point v. 4 Stratification of Codimension-One Manifolds For codimension-one varieties, the normal space at a smooth point is spanned by a unique vector (up to scaling) and hence is one dimensional. Consequently, the t… view at source ↗
Figure 7
Figure 7. Figure 7: Partition of torus X when the unit ball is the cube as in Theorem 4.7. X0 is the set of points in light blue, X1 is the set of points in orange, and X2 is the set of points in purple. Example 4.7. The transversality hypothesis in Theorem 4.4 is necessary to guarantee that XB is a stratifica￾tion of X. As an example of the behavior that can arise when this hypothesis is not satisfied, consider when B is the… view at source ↗
Figure 8
Figure 8. Figure 8: Stratification of the curves in Theorem 3.9. For each curve Vi , i = 1, 2, 3, 4, the stratum (Vi)0 is the set of points depicted in blue, and the stratum (Vi)1 is the set of points depicted in red. Example 4.10. Consider the hyperboloid X from Theorem 3.14 with the norm induced by the cube. The stratification given by Theorem 3.6 is shown in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Stratification of the hyperboloid X = V(36x 2+9y 2−4z 2−36) induced by the polyhedral norm with the cube as its unit ball. The strata X0, X1, and X2 are depicted in orange, black, and blue, respectively. Remark 4.11. It is possible that for a variety X, some of the strata may be empty. This is the case for hyperplanes in R n. Moreover, [1, Theorem 5.1] gives an upper bound on the number of full-dimensional… view at source ↗
Figure 10
Figure 10. Figure 10: The variety X from Theorem 5.4 with full-dimensional medial axis. Theorems 5.2 to 5.4 show that the medial axis of a variety X ⊂ R n can exhibit a range of different behaviors. In particular, it can be empty, full-dimensional, or contain components of different dimensions. In what follows, we focus on finding an algebraic set that contains the medial axis of a codimension-one variety X = V(f) for some pol… view at source ↗
Figure 11
Figure 11. Figure 11: Consider the variety X = V(x 2 + y 2 − 1) ⊂ R 2 with respect to polyhedral norm with unit ball given by the square. Each figure represents the algebraic closure of the locus of equidistant points arising from different pairs of faces. The variety X is plotted in all the pictures for reference. Similarly, the ellipses in Figure 11c are generated by considering pairs of faces (F1, F2) where F1 is a vertex a… view at source ↗
read the original abstract

We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the point and the inner normal fan of the polyhedral ball. For codimension-one varieties, we decompose $X$ into sets of points whose Voronoi cones have the same dimension, which is the expected dimension of their Voronoi cell. We prove that this decomposition is a stratification of $X$ and that each strata is a semialgebraic set. We conclude by giving an algebraic description of the medial axis, which is the locus of points whose minimal distance to $X$ is achieved at more than one point on $X$.

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 studies distance minimization from points in R^n to a real algebraic variety X under a polyhedral norm. For codimension-one varieties, it decomposes X according to the dimension of the Voronoi cone at each point (determined by the normal space and the inner normal fan of the polyhedral ball), proves that the resulting decomposition is a stratification whose strata are semialgebraic, and supplies an algebraic description of the medial axis (the set of points whose minimal distance to X is realized at more than one point of X).

Significance. If the claims hold, the work supplies a semialgebraic stratification of codimension-one varieties that organizes Voronoi cells in polyhedral norms and yields an algebraic characterization of the medial axis. This framework connects real algebraic geometry with convex geometry and could support symbolic or computational methods for optimization and medial-axis computations over varieties equipped with non-Euclidean norms. The explicit use of stratification and semialgebraicity is a methodological strength.

minor comments (3)
  1. Abstract: the phrase 'each strata is a semialgebraic set' is grammatically incorrect and should read 'each stratum is a semialgebraic set'.
  2. The introduction would benefit from a short paragraph contrasting the polyhedral-norm setting with the classical Euclidean case to clarify the technical novelty.
  3. Notation for the inner normal fan and Voronoi cone should be introduced with a small illustrative example (e.g., the L1 or L-infinity ball) to aid readability for readers outside convex geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive evaluation of its significance. The recommendation for minor revision is noted; we will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a stratification of a codimension-one real algebraic variety X by grouping points according to the dimension of their Voronoi cones (determined by the normal space and the inner normal fan of a polyhedral ball), proves the strata are semialgebraic, and gives an algebraic description of the medial axis. These steps rely on standard external notions from real algebraic geometry (semialgebraic sets, stratifications) and convex geometry (Voronoi cells, polyhedral norms), with no equations or definitions that reduce the claimed results to fitted parameters, self-referential constructions, or load-bearing self-citations. The abstract and structure indicate an independent proof chain grounded in prior mathematical facts outside the paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background from algebraic geometry and convex geometry; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Real algebraic varieties are zero sets of polynomials over the reals.
    Basic definition invoked when referring to X as a real algebraic variety.
  • domain assumption Polyhedral norms induce Voronoi cells whose geometry is determined by the normal space and the inner normal fan of the unit ball.
    Core geometric assumption stated in the abstract for the distance minimization problem.

pith-pipeline@v0.9.0 · 5436 in / 1211 out tokens · 40391 ms · 2026-05-10T01:58:34.875171+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

15 extracted references · 15 canonical work pages

  1. [1]

    Voronoi diagrams of algebraic varieties under polyhedral norms.J

    Adrian Becedas, Kathl´ en Kohn, and Lorenzo Venturello. Voronoi diagrams of algebraic varieties under polyhedral norms.J. Symbolic Comput., 120:Paper No. 102229, 14, 2024

    work page 2024

  2. [2]

    General topology

    Nicolas Bourbaki.Elements of mathematics. General topology. Part 1. Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966

    work page 1966

  3. [3]

    Birkh¨ auser/Springer, Cham, 2024

    Paul Breiding, Kathl´ en Kohn, and Bernd Sturmfels.Metric algebraic geometry, volume 53 ofOberwol- fach Seminars. Birkh¨ auser/Springer, Cham, 2024

    work page 2024

  4. [4]

    Euclidean distance degree and mixed volume

    Paul Breiding, Frank Sottile, and James Woodcock. Euclidean distance degree and mixed volume. Found. Comput. Math., 22(6):1743–1765, 2022. 20

    work page 2022

  5. [5]

    Optimal transport to a variety

    T¨ urk¨ u¨Ozl¨ um C ¸ elik, Asgar Jamneshan, Guido Mont´ ufar, Bernd Sturmfels, and Lorenzo Venturello. Optimal transport to a variety. InMathematical aspects of computer and information sciences, volume 11989 ofLecture Notes in Comput. Sci., pages 364–381. Springer, Cham, 2020

    work page 2020

  6. [6]

    Wasserstein distance to independence models.J

    T¨ urk¨ u¨Ozl¨ um C ¸ elik, Asgar Jamneshan, Guido Mont´ ufar, Bernd Sturmfels, and Lorenzo Venturello. Wasserstein distance to independence models.J. Symbolic Comput., 104:855–873, 2021

    work page 2021

  7. [7]

    Voronoi cells of vari- eties.J

    Diego Cifuentes, Kristian Ranestad, Bernd Sturmfels, and Madeleine Weinstein. Voronoi cells of vari- eties.J. Symbolic Comput., 109:351–366, 2022

    work page 2022

  8. [8]

    Tropical bisectors and Voronoi diagrams

    Francisco Criado, Michael Joswig, and Francisco Santos. Tropical bisectors and Voronoi diagrams. Found. Comput. Math., 22(6):1923–1960, 2022

    work page 1923

  9. [9]

    De Loera, J¨ org Rambau, and Francisco Santos.Triangulations, volume 25 ofAlgorithms and Computation in Mathematics

    Jes´ us A. De Loera, J¨ org Rambau, and Francisco Santos.Triangulations, volume 25 ofAlgorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2010. Structures for algorithms and applications

    work page 2010

  10. [10]

    Jan Draisma, Emil Horobet ¸, Giorgio Ottaviani, Bernd Sturmfels, and Rekha R. Thomas. The Euclidean distance degree of an algebraic variety.Found. Comput. Math., 16(1):99–149, 2016

    work page 2016

  11. [11]

    Polyhedral combinatorics of bisectors.Adv

    Aryaman Jal and Katharina Jochemko. Polyhedral combinatorics of bisectors.Adv. Geom., 25(2):147– 174, 2025

    work page 2025

  12. [12]

    Lee.Introduction to topological manifolds, volume 202 ofGraduate Texts in Mathematics

    John M. Lee.Introduction to topological manifolds, volume 202 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2000

    work page 2000

  13. [13]

    Lee.Introduction to smooth manifolds, volume 218 ofGraduate Texts in Mathematics

    John M. Lee.Introduction to smooth manifolds, volume 218 ofGraduate Texts in Mathematics. Springer, New York, second edition, 2013

    work page 2013

  14. [14]

    Maxim, Jose I

    Laurentiu G. Maxim, Jose I. Rodriguez, and Botong Wang. Euclidean distance degree of the multiview variety.SIAM J. Appl. Algebra Geom., 4(1):28–48, 2020

    work page 2020

  15. [15]

    Ziegler.Lectures on polytopes, volume 152 ofGraduate Texts in Mathematics

    G¨ unter M. Ziegler.Lectures on polytopes, volume 152 ofGraduate Texts in Mathematics. Springer- Verlag, New York, 1995. Authors’ addresses: Eliana Duarte, Universidade do Portoeliana.gelvez@fc.up.pt Nidhi Kaihnsa, University of Copenhagennidhi@math.ku.dk Julia Lindberg, Georgia Institute of Technologyjlindberg3@math.gatech.edu Ang´ elica Torres, Universi...

    work page 1995