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arxiv: 2304.08626 · v2 · submitted 2023-04-17 · 🧮 math.OC · cs.DM

On taxicab distance mean functions and their geometric applications: methods, implementations and examples

Csaba Vincze , \'Abris Nagy This is my paper

Pith reviewed 2026-05-24 09:03 UTC · model grok-4.3

classification 🧮 math.OC cs.DM
keywords taxicab distancedistance mean functionsgeneralized conicsgeometric tomographycoordinate X-raysfocal set bisectionreconstruction problems
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The pith

Taxicab distance mean functions produce generalized conics that bisect focal sets and reconstruct objects from coordinate X-rays.

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

The paper surveys taxicab distance mean functions, which compute the average distance from a point to a focal set of points, with their level sets termed generalized conics. When the focal set is infinite, the average is obtained by integration over that set. These objects are then applied to two tasks in geometric tomography: bisecting the focal set itself and recovering an object from its coordinate X-ray projections. Concrete Maple implementations and examples demonstrate the methods. A reader would care because the constructions turn average-distance level sets into practical tools for tomography problems that are otherwise handled by more complex integral transforms.

Core claim

Taxicab distance mean functions, defined either by summation over finite focal points or integration over infinite focal sets, generate generalized conics whose geometric properties directly support bisection of the focal set and solution of reconstruction problems from coordinate X-rays in geometric tomography.

What carries the argument

Taxicab distance mean function, which averages distance to a focal set and whose level sets are generalized conics used for tomography tasks.

If this is right

  • Bisection of any focal set becomes feasible once its taxicab distance mean function is known.
  • Reconstruction from coordinate X-rays reduces to locating appropriate level sets of the mean function.
  • Maple implementations supply explicit algorithms that realize both bisection and reconstruction for chosen focal sets.
  • The same level-set geometry applies uniformly to both finite and infinite focal sets.

Where Pith is reading between the lines

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

  • The integration step for infinite sets may allow the same framework to handle continuous distributions of focal points beyond the discrete cases shown.
  • Coordinate X-ray reconstruction via these means could be compared directly with classical Radon-transform methods to measure computational savings.
  • The bisection property might extend to other Minkowski norms if the level-set geometry remains sufficiently rigid.

Load-bearing premise

The average distance to an infinite focal set is obtained by integration over that set, so that the resulting level sets behave as required for bisection and X-ray reconstruction.

What would settle it

A concrete focal set for which the taxicab mean level sets fail to produce a valid bisection or yield incorrect reconstructions from coordinate X-ray data would disprove the claimed applications.

Figures

Figures reproduced from arXiv: 2304.08626 by \'Abris Nagy, Csaba Vincze.

Figure 1
Figure 1. Figure 1: The sequence of points Xk generated by the above procedure for k = 1, 2, . . . 50. Darker points present elements Xk with higher indices k. Notice how these sequences of points converge to the minimizer of the taxicab distance mean function (5). Then the stochastic algorithm for finding the minimizer of the taxicab distance mean function (5) of a polygon can be implemented in Maple as follows, see [PITH_F… view at source ↗
Figure 2
Figure 2. Figure 2: X-rays of compact planar bodies. 3. Applications in geometric tomography The unweighted function (5) is strongly related to the parallel X-rays as follows: by the Cavalieri ´ principle, formula DifK(x) = µn(K ≤i x i ) − µn(x i ≤i K) (i = 1, . . . , n) of the first partial derivatives implies that DiDifK(x) =a.e. 2XiK(x i ) (i = 1, . . . , n), (12) where XiK(x i ) := µn−1(x i =i K) is the (n − 1)-dimensiona… view at source ↗
Figure 3
Figure 3. Figure 3: The set we are looking for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The coordinate X-rays [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The optimal solution under low resolution (left) and a greedy version under high resolution (right). [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A switching chain of 10 elements [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.

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 / 2 minor

Summary. The manuscript surveys taxicab (L1) distance mean functions, whose level sets are generalized conics. For finite focal sets the mean is the ordinary average; for infinite focal sets it is defined by integration over the focal set. The survey focuses on applications in geometric tomography, specifically bisection of the focal set and reconstruction from coordinate X-rays, and illustrates the results with Maple implementations, explicit methods, and numerical examples.

Significance. If the integration step produces well-defined level sets that inherit the geometric properties needed for tomography, the compilation of theoretical results together with reproducible Maple code supplies a practical reference for L1-metric methods in geometric tomography. The provision of concrete implementations is a positive feature that lowers the barrier to applying these constructions.

major comments (1)
  1. [§2] §2 (definition of the mean for continuous focal sets): the manuscript states that the average distance 'is typically given by integration over the focal set' but supplies neither a proof that the integral is finite and parametrization-independent nor a verification that the resulting sublevel sets satisfy the bisection or injectivity properties required for coordinate X-ray reconstruction. This step is load-bearing for the tomography claims.
minor comments (2)
  1. [Abstract] The abstract and introduction use the phrase 'typically' without citing the precise conditions under which the integral definition is known to be valid; a short clarifying sentence with references would improve readability.
  2. [Examples section] Figure captions for the Maple-generated examples should explicitly state the focal set, the measure used for integration, and the coordinate system to allow direct reproduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting the need for additional justification in the definition of the mean for continuous focal sets. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§2] §2 (definition of the mean for continuous focal sets): the manuscript states that the average distance 'is typically given by integration over the focal set' but supplies neither a proof that the integral is finite and parametrization-independent nor a verification that the resulting sublevel sets satisfy the bisection or injectivity properties required for coordinate X-ray reconstruction. This step is load-bearing for the tomography claims.

    Authors: The manuscript is a survey that presents the integration-based definition for infinite focal sets as the standard construction in the literature on generalized conics. We acknowledge that an explicit verification of finiteness, parametrization independence, and the inheritance of bisection/injectivity properties is not supplied in §2 and is indeed load-bearing for the tomography applications. In the revised version we will add a short paragraph in §2 supplying (i) a direct argument that the L1 integral remains finite on compact focal sets (by boundedness of the taxicab distance) and is independent of the chosen parametrization (by the change-of-variable formula for rectifiable curves), and (ii) a reference to the fact that the bisection property is preserved because the mean is a convex combination (or integral) of convex functions, together with a citation to the injectivity results already established for coordinate X-rays of L1 level sets in the cited tomography literature. These additions will be kept concise so as not to alter the survey character of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; survey relies on external literature for definitions.

full rationale

The paper is explicitly a survey on applications of taxicab distance mean functions and generalized conics in geometric tomography. The abstract states that for infinite focal sets the average distance 'is typically given by integration over the focal set,' which is presented as a standard definition drawn from prior literature rather than a result derived or fitted within this manuscript. No equations, self-citations, or ansatzes are shown to reduce the central claims (bisection and X-ray reconstruction) to inputs by construction. The derivation chain is self-contained against external benchmarks, with implementations in Maple serving as illustrations rather than load-bearing proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard concepts from metric geometry and tomography without introducing new fitted parameters or invented entities visible in the abstract.

axioms (2)
  • standard math Taxicab metric satisfies the properties of a normed vector space
    Core background assumption for defining distance mean functions.
  • domain assumption Level sets of distance mean functions are generalized conics
    Definition invoked for the geometric applications.

pith-pipeline@v0.9.0 · 5623 in / 1081 out tokens · 28400 ms · 2026-05-24T09:03:32.076400+00:00 · methodology

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Reference graph

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