Double Spherical Cap Rearrangement

• Double spherical cap rearrangement is a process that redistributes each circular slice of a planar set into two diametrically symmetric arcs while preserving overall area and the barycenter.
• It guarantees a perimeter non-increasing transformation under the specific condition that non-trivial slices are disconnected, establishing a sharp isoperimetric result in the plane.
• The method distinguishes itself from classical symmetrizations by its barycenter preservation and reveals limitations in higher dimensions through explicit counterexamples.

Double spherical cap rearrangement is a symmetrization procedure for planar sets that redistributes each circular slice of a set into two diametrically symmetric arcs while preserving the slice length, and hence the total area; in the planar formulation developed in "The double spherical cap rearrangement of planar sets" (Gambicchia, 17 Sep 2025), it also preserves the barycenter when centered at the barycenter. The principal result is an isoperimetric one: under the assumption that all non-trivial spherical slices are disconnected, the rearrangement does not increase perimeter. The same source also shows that the analogous perimeter monotonicity fails in dimensions $N \geq 3$ by an explicit counterexample. Related work on spherical Steiner symmetrization situates this rearrangement within a broader family of cap-based symmetrizations that reduce geometric problems to spherical caps (Basit et al., 2024), while the separate double cap conjecture concerns orthogonality-avoiding subsets of the sphere rather than rearrangements of planar sets (Czifra et al., 27 May 2026).

1. Geometric definition

Let $E \subset \mathbb{R}^2$ be a set of finite perimeter and area. For each radius $r>0$, the corresponding spherical slice is

$E_r := E \cap \partial B(r),$

that is, the intersection of $E$ with the circle centered at the origin of radius $r$. Its circular distribution is

$v(r) := \mathcal{H}^1(E_r),$

where $\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. The rearrangement is defined slice by slice from this radial data (Gambicchia, 17 Sep 2025).

For each $r>0$, one sets

$E \subset \mathbb{R}^2$0

so that the total length $E \subset \mathbb{R}^2$1 is redistributed into two diametrically symmetric arcs of length $E \subset \mathbb{R}^2$2 each. The double cap at level $E \subset \mathbb{R}^2$3 is

$E \subset \mathbb{R}^2$4

where $E \subset \mathbb{R}^2$5 denotes the angle associated to $E \subset \mathbb{R}^2$6. The double spherical cap rearrangement is then

$E \subset \mathbb{R}^2$7

This construction is $E \subset \mathbb{R}^2$8-symmetric, specifically symmetric with respect to the $E \subset \mathbb{R}^2$9-axis. The source states that it preserves both the area and, crucially, the barycenter of the original set when centered at the barycenter (Gambicchia, 17 Sep 2025). This identifies the rearrangement as a barycentrically adapted symmetrization rather than a purely radial one.

2. Slice structure and the role of disconnectedness

The central structural notion is the set of single-arched radii,

$r>0$0

These are precisely the radii at which the slice is a single non-trivial arc rather than several disjoint arcs or a full circle (Gambicchia, 17 Sep 2025).

The main planar perimeter statement is expressed locally in polar annuli. For any Borel set $r>0$1,

$r>0$2

where $r>0$3 denotes the perimeter and $r>0$4 are polar coordinates. In particular, if all non-trivial slices are disconnected, equivalently if $r>0$5, then

$r>0$6

The source explains the geometric reason for the exceptional set $r>0$7. At radii in $r>0$8, the rearrangement splits a single arc into two arcs, thereby introducing new boundary and potentially increasing perimeter. Outside this set, the rearrangement is at least as efficient as the original configuration in perimeter terms. This makes disconnectedness of non-trivial spherical slices the decisive hypothesis for the planar isoperimetric property (Gambicchia, 17 Sep 2025).

3. Isoperimetric property in the plane

The planar theorem establishes that the double spherical cap rearrangement is perimeter non-increasing under the disconnection assumption on non-trivial slices. The result is formulated not only globally but also as a localized inequality on $r>0$9, which encodes radial shells and permits fine control of the perimeter contribution as a function of $E_r := E \cap \partial B(r),$0 (Gambicchia, 17 Sep 2025).

The proof, as summarized in the source, proceeds through geometric measure theory and BV techniques. Factually, the construction preserves the length of each slice $E_r := E \cap \partial B(r),$1 while replacing its angular distribution by two symmetric caps; the analysis then compares the resulting boundary geometry to that of the original set. The statement that the perimeter is “typically (except for certain slices) strictly decreased” appears in the abstract-level summary, while the quantitative theorem records the exact defect term $E_r := E \cap \partial B(r),$2.

This suggests that the rearrangement is not a universal perimeter minimizer among sets with the same circular distribution, but rather a sharp symmetrization for a class determined by slice topology. A plausible implication is that the theorem isolates the precise obstruction to monotonicity in two dimensions: connected non-trivial slices.

4. Preservation principles and comparison with other symmetrizations

The double spherical cap rearrangement is presented as distinct from more standard spherical rearrangements because it preserves not only area or volume but also the barycenter, a feature identified as vital in problems with barycentric constraints (Gambicchia, 17 Sep 2025). The source contrasts this with Steiner and spherical rearrangements, which produce symmetric sets but do not in general preserve the barycenter.

Within the broader symmetrization literature, "Steiner symmetrization on the sphere" introduces a different spherical symmetrization acting on subsets of $E_r := E \cap \partial B(r),$3 rather than on planar sets decomposed by Euclidean circles (Basit et al., 2024). That work shows that spherical Steiner symmetrization preserves spherical volume in every dimension, preserves convexity in the spherical plane under an angular monotonicity property, and does not generally preserve convexity in dimensions $E_r := E \cap \partial B(r),$4. It also proves perimeter and diameter monotonicity in $E_r := E \cap \partial B(r),$5 under the same angular monotonicity condition and derives convergence to a spherical cap for centrally symmetric convex disks or convex disks of diameter $E_r := E \cap \partial B(r),$6.

The relation between the two constructions is therefore methodological rather than literal. Both are cap-based symmetrizations, both preserve a measure quantity slice by slice, and both support reductions to spherical caps in geometric inequalities. However, the double spherical cap rearrangement of planar sets is tailored to Euclidean planar sets and barycentric constraints, whereas spherical Steiner symmetrization is formulated intrinsically on the sphere (Basit et al., 2024).

5. Failure in higher dimensions

The higher-dimensional analogue does not satisfy the same perimeter monotonicity. The planar source gives an explicit counterexample for $E_r := E \cap \partial B(r),$7, thereby explaining why an analogous result cannot hold in general (Gambicchia, 17 Sep 2025).

The construction consists of a ball centered at the origin together with two thin cylinders of equal length $E_r := E \cap \partial B(r),$8 but different section-radii $E_r := E \cap \partial B(r),$9 and $E$0, with $E$1, attached symmetrically to the ball. After double spherical cap rearrangement, one obtains a ball with two cylinders of length $E$2 and radius $E$3 determined by

$E$4

The perimeter comparison, up to small error terms, is

$E$5

$E$6

Because the function $E$7 is concave for $E$8, the rearranged perimeter is greater: $E$9

This counterexample shows that disconnectedness of slices is no longer sufficient in higher dimensions. In the planar theory, counting endpoints on circular slices is decisive; in higher dimensions, the geometry of cap sections interacts with concavity in a way that reverses the perimeter comparison (Gambicchia, 17 Sep 2025).

6. Relation to spherical caps, extremal problems, and the double cap conjecture

Cap-based rearrangements are a recurring mechanism in isoperimetric analysis. The spherical Steiner framework explicitly states that symmetrization “drives sets towards spherical caps” under symmetry or diameter hypotheses, and uses this to prove spherical analogues of theorems of Sas and Winternitz and to confirm a conjecture of Besau and Werner for centrally symmetric spherically convex disks (Basit et al., 2024). The same source states that this matches the philosophy of the double spherical cap rearrangement: for isoperimetric questions on $r$0, symmetrization methods reduce general convex sets to symmetric arrangements, namely caps.

A separate but terminologically related development is the double cap conjecture in extremal combinatorial geometry. There, one studies the maximum possible density $r$1 of a measurable subset $r$2 containing no pair of orthogonal vectors, and the “double cap” refers to the union of two open antipodal spherical caps of radius $r$3 (Czifra et al., 27 May 2026). For $r$4, this construction gives density $r$5, conjectured by Gil Kalai to be optimal for all $r$6. The 2026 work improves the upper bound for $r$7 to $r$8, from the previous $r$9, using harmonic-analytic arguments and the geometric fractional chromatic number of finite graphs (Czifra et al., 27 May 2026).

These notions should not be conflated. In the conjectural extremal problem, the double cap is a candidate maximizer for density under an orthogonality-avoidance constraint. In the planar rearrangement problem, the double spherical cap rearrangement is a deterministic slice-wise transformation of sets. The shared terminology reflects the common geometric role of antipodal caps, but the analytic settings, admissible objects, and optimization criteria are different.

7. Mathematical significance and limitations

The significance assigned to the double spherical cap rearrangement is tied to barycentric isoperimetric inequalities and related variational problems in which the location of the set cannot be altered (Gambicchia, 17 Sep 2025). Because the rearrangement preserves both area and barycenter, it is adapted to settings where comparison with centered balls or symmetric competitors must respect a fixed center of mass.

Its scope is simultaneously sharp and restricted. In the plane, the theorem characterizes when the rearrangement is safe for perimeter reduction or control: exactly when non-trivial slices are disconnected. In higher dimensions, even under disconnectedness of slices, perimeter can increase. This suggests that the planar statement is not merely a first case of a dimension-free principle, but a genuinely two-dimensional phenomenon tied to the geometry of arcs on circles and the endpoint structure of slices.

The resulting picture is therefore stratified. For planar sets, double spherical cap rearrangement is a barycenter-preserving symmetrization with a precise isoperimetric inequality under a topological hypothesis on slices (Gambicchia, 17 Sep 2025). For intrinsic spherical geometry, related cap-based symmetrizations support convergence and extremality results on $v(r) := \mathcal{H}^1(E_r),$0 (Basit et al., 2024). For orthogonality-avoiding sets on the sphere, double caps arise instead as extremal candidates in a density problem whose best known bounds remain close but not equal in dimension three (Czifra et al., 27 May 2026).