The mathematical analysis shows that a completely flattened “needle” can attain the best possible value in several weighted-perimeter problems for symmetric convex curves in the plane. The result is exact within the stated class, but it does not establish that needles—or flat, disk-like bodies—are optimal for every convex shape or every dimension.
The work studies convex bodies and curves rather than an empirical sample. It asks how a radial weighted perimeter—a boundary measure that weights points according to their distance from the origin—changes when ordinary perimeter is held fixed, and when a degenerate configuration can be an optimizer.
What the analysis establishes
For every positive fixed perimeter, the analysis proves that minimizers exist for increasing radial weights and maximizers exist for decreasing ones, within the paper’s specified class of convex bodies and admissible weights. The proof uses compactness, showing that suitable sequences of shapes retain the structure needed to reach an extremum.
The authors also rule out degenerate extremal bodies in some parameter ranges. The paper establishes nonempty interior for the E_p minimizer when p is greater than max{4 − n, 0}, and for the E_−α maximizer when n is at least 5 and 0 < α < n − 4. These conclusions come from continuous perturbations that preserve perimeter while giving the shape cylindrical thickness.
Another result places the origin on the boundary of an extremal body when the radial weight has the required smoothness and a positive Laplacian. For the inverse-power functional E_−α, the stated range is n − 2 < α < n − 1.
Why the planar cases stand out
In the main planar symmetry class—curves with two orthogonal axes meeting at the origin—the authors derive an upper bound for convex decreasing weights. In their notation, the weighted perimeter is no greater than 4∫₀^{|Γ|/4}F(t²)dt, where |Γ| is the ordinary perimeter and F is the weight expressed as a function of squared distance.
For concave increasing weights, the inequality reverses. A needle reaches equality in both planar inequalities, and when the relevant convexity or concavity is strict, it is the only equality shape in the corresponding case.
The paper spells out three examples. For power weights, it gives E_p(Γ) ≥ |Γ|^(p+1)/((p+1)4^p) for 0 < p ≤ 2, with needle equality stated for p < 2. It also gives a logarithmic upper bound, E_log(Γ) ≤ |Γ|(1 − log(|Γ|/4)), and an inverse-power upper bound, E_−α(Γ) ≤ 4^α|Γ|^(1−α)/(1−α) for 0 < α < 1; the text states needle equality for both of those bounds.
The proof combines rearrangement with a continuous form of Karamata’s majorization inequality, a way of comparing how quantities are distributed. The planar inequalities also extend beyond the two-axis-symmetry setting to 4-quadrant-convex curves, although that remains a restricted class rather than all convex curves.
The larger question is still open
The planar needle results support, but do not prove, the authors’ conjecture that for n − 2 < α < n − 1 the E_−α maximizer in R^n is degenerate and specifically takes the form of a flat ball—a disk-like body with no thickness in one direction. The full conjecture has not been established without the relevant symmetry restrictions.
The paper reports no result supporting or discouraging the conjecture in the range n − 4 ≤ α ≤ n − 2. Its nondegeneracy conclusions also cover only the parameter ranges stated above, leaving other ranges unresolved.
A separate necessary condition says that any body critical for E_p under a fixed perimeter, when p > 2 − n, must have zero p-weighted first moment: ∫∂Ω|x|^(p−2)x dH^(n−1)(x) = 0. That condition identifies a requirement for criticality, not a complete description of the optimizer.
The document is an arXiv preprint, version 1, dated 20 August 2026. It contains theorem-level mathematical analysis of admissible shapes, not an empirical dataset.
Paper data and sources
Original title: Weighted Perimeters and pth Moments of Inertia of Convex Curves and Surfaces
Authors: Gyula Csató, Davide Giovagnoli, Prosenjit Roy
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text