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arxiv: 2605.17484 · v1 · pith:C4ZIP7D5new · submitted 2026-05-17 · 🧮 math.AP

On the Stability of Inverse Conductivity Problem for Polyhedral Inclusions under a Single Measurement

Chun-Hsiang Tsou This is my paper

Pith reviewed 2026-05-19 22:23 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse conductivity problemstability estimatepolyhedral inclusionHausdorff distancesingle measurementlogarithmic stabilitysingularity decompositionmicrolocal analysis
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The pith

Logarithmic stability estimate holds for the Hausdorff distance between convex polyhedral inclusions from a single boundary measurement error.

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

The paper establishes a logarithmic stability estimate for the Hausdorff distance between two convex polyhedral inclusions embedded in a homogeneous isotropic medium, expressed in terms of the error in a single boundary measurement. The proof combines singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis. A sympathetic reader would care because the result quantifies how measurement noise limits the recoverable accuracy of the inclusion shape under minimal data, which matters for applications where only one experiment is feasible.

Core claim

Combining singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis, we establish a logarithmic stability estimate for the Hausdorff distance between inclusions in terms of the measurement error.

What carries the argument

Singularity decomposition near the edges and vertices of the convex polyhedron, which produces explicit leading terms in the solution expansion that enable propagation of smallness estimates to bound the geometric difference.

If this is right

  • The Hausdorff distance between two such inclusions is bounded by a constant times the logarithm of the measurement error.
  • A single boundary measurement suffices to obtain quantitative stability for the location and shape of the inclusion.
  • The estimate relies on the geometric singularities at edges and vertices that are characteristic of convex polyhedra.

Where Pith is reading between the lines

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

  • The same combination of tools might yield analogous logarithmic stability for polyhedral inclusions in other elliptic inverse problems with limited data.
  • Numerical tests could add controlled noise to simulated measurements and check whether the observed Hausdorff error grows no faster than the logarithmic bound.
  • The result suggests that convexity and polyhedral structure are key to achieving stability from minimal measurements, raising the question of how far the assumption can be relaxed while preserving the singularity-driven argument.

Load-bearing premise

The unknown inclusion is a convex polyhedron and the background medium is homogeneous and isotropic, so that the solution exhibits a specific singularity structure near edges and vertices.

What would settle it

Exhibit two distinct convex polyhedra whose single boundary measurements differ by an arbitrarily small amount yet whose Hausdorff distance exceeds any multiple of the logarithm of that measurement difference.

Figures

Figures reproduced from arXiv: 2605.17484 by Chun-Hsiang Tsou.

Figure 1
Figure 1. Figure 1: Polyhedral corner at vertex v. E, z v O ρ F+ F− Π xˆ yˆ a [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Local cylindrical coordinates (r, θ, z) around the edge E. The origin O lies on E at distance ρ from the vertex v. The plane Π is perpen￾dicular to E at O; the two faces F± meet along E with opening angle a. • Γ± := ∂D ∩ Π denote the two sides of the corner at O in the plane Π. In local cylindrical coordinates, we impose that Γ± = {(r, θ, 0) | 0 < r < h, θ = ±a/2} . • From the choice made in the previous p… view at source ↗
Figure 3
Figure 3. Figure 3: Cross-section in the plane Π perpendicular to E at O. The two sides Γ± (thick black) enclose the opening angle a. The contour ∂Se (red) lies at distance 1/τ from Γ± on the exterior side, ∂Si (blue) is an arc on the circle r = h, and the shaded region is A−. 4.2. Integral Identity and Estimates. The following integral identity is the key ingre￾dient for our stability estimate. Proposition 4.1. Let u0 ∈ H1 l… view at source ↗
read the original abstract

In this paper, we study the stability of the inverse conductivity problem of determining a convex polyhedral inclusion embedded in a homogeneous isotropic medium from a single boundary measurement. The main tools in our analysis are singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis. Combining these tools, we establish a logarithmic stability estimate for the Hausdorff distance between inclusions in terms of the measurement error.

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 studies the stability of the inverse conductivity problem of recovering a convex polyhedral inclusion in a homogeneous isotropic medium from a single boundary measurement. Combining singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis, the authors derive a logarithmic stability estimate relating the Hausdorff distance between two such inclusions to the measurement error.

Significance. If the result holds with uniform constants, it would provide a useful quantitative stability bound for an inverse problem under minimal data, which is relevant for applications. The direct analytic approach via established microlocal and singularity techniques is a strength, avoiding parameter fitting or self-referential definitions.

major comments (2)
  1. [§3] §3 (singularity decomposition near vertices/edges): the leading singular coefficients depend on the dihedral angles and the number of faces meeting at each vertex. The proof must establish that the map from these coefficients to the Hausdorff distance yields constants independent of the specific geometry within the class of convex polyhedra; otherwise the stability estimate is not uniform over the admissible set.
  2. [Theorem 1.1 / §5] Main stability estimate (Theorem 1.1 or equivalent in §5): the logarithmic rate d_H(K1,K2) ≲ |log ε|^{-α} relies on the difference in singular expansions controlling the geometry uniformly. If the implicit constant deteriorates as an angle approaches 0 or π, or as the number of vertices grows, this step is load-bearing and requires explicit verification or a counter-example exclusion.
minor comments (2)
  1. [Abstract] The abstract should specify the precise norm (e.g., L^2(∂Ω) or H^{-1/2}(∂Ω)) in which the measurement error ε is measured.
  2. [§1] Notation for the conductivity equation and the admissible class of polyhedra (e.g., bounds on number of faces or minimal angle) could be introduced earlier for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (singularity decomposition near vertices/edges): the leading singular coefficients depend on the dihedral angles and the number of faces meeting at each vertex. The proof must establish that the map from these coefficients to the Hausdorff distance yields constants independent of the specific geometry within the class of convex polyhedra; otherwise the stability estimate is not uniform over the admissible set.

    Authors: We agree that the leading singular coefficients in the decomposition depend on the local geometry, specifically the dihedral angles and the number of faces meeting at vertices. Our analysis in §3 derives these coefficients via the standard theory for elliptic equations in polyhedral domains and then propagates the information to the Hausdorff distance via quantitative unique continuation. The constants remain uniform over the admissible class because all estimates are taken with respect to a fixed bounded domain containing the inclusions and a fixed conductivity contrast; convexity prevents degeneracy that would make the constants blow up. To address the concern explicitly, we will add a clarifying remark in the revised §3 stating that the implicit constants depend only on these global a priori bounds and not on the specific angles or vertex counts within the convex polyhedral class. revision: partial

  2. Referee: [Theorem 1.1 / §5] Main stability estimate (Theorem 1.1 or equivalent in §5): the logarithmic rate d_H(K1,K2) ≲ |log ε|^{-α} relies on the difference in singular expansions controlling the geometry uniformly. If the implicit constant deteriorates as an angle approaches 0 or π, or as the number of vertices grows, this step is load-bearing and requires explicit verification or a counter-example exclusion.

    Authors: The logarithmic stability in Theorem 1.1 follows from controlling the difference of the singular expansions and then applying propagation of smallness. The implicit constants are independent of the specific geometry because the microlocal estimates and the smallness propagation are performed uniformly for all convex polyhedra inside the fixed domain; the worst-case bounds from elliptic regularity already account for angles in (0,π) and any finite number of vertices. We do not see deterioration within the admissible class, as convexity and the fixed ambient domain prevent the constants from becoming arbitrarily large. We will insert a short paragraph in §5 making this uniformity explicit and noting that the result holds with constants depending only on the a priori data. revision: partial

Circularity Check

0 steps flagged

No circularity: direct analytic proof from established tools

full rationale

The paper derives a logarithmic stability estimate for the Hausdorff distance between convex polyhedral inclusions by combining singularity decomposition for elliptic equations in non-smooth domains, propagation of smallness, and microlocal analysis. These are standard, externally established techniques whose validity does not depend on the target result or on any fitted parameters internal to this work. The abstract and claimed derivation chain contain no self-definitional steps, no renaming of known results as new predictions, and no load-bearing self-citations that reduce the central claim to an unverified prior result by the same authors. The argument is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger is therefore populated from the stated assumptions and tools rather than from detailed derivations.

axioms (2)
  • standard math Singularity decomposition holds for elliptic equations in domains with polyhedral boundaries
    Invoked to separate singular and regular parts of the solution near edges and vertices.
  • standard math Propagation of smallness applies to solutions of the conductivity equation
    Used to control interior behavior from boundary data.

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