Pith Number

pith:3IEJ2MFY

pith:2026:3IEJ2MFYEJ5GEHBKUYYFEO3P2W

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The lens cluster and triod cluster uniquely minimize the anisotropic perimeter in $\mathbb{R}^2$

Paula Benitez

For regular anisotropies the only local minimizers of the perimeter among (1,2)- and (1,3)-clusters in the plane are the standard lens and triod shapes.

arxiv:2604.03510 v2 · 2026-04-03 · math.AP

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Claims

C1strongest claim

for regular (smooth, symmetric, and uniformly convex) anisotropies, we prove that a cluster is a local minimizer if and only if, up to translations, it is a standard anisotropic lens cluster in the (1,2)-cluster case, or a standard anisotropic triod cluster in the (1,3)-cluster case.

C2weakest assumption

The anisotropy is assumed to be regular, i.e., smooth, symmetric, and uniformly convex; the proof of uniqueness and the approximation argument to general anisotropies both rely on this regularity (abstract, paragraph beginning 'Our main results provide a geometric characterization').

C3one line summary

For regular anisotropies the lens cluster uniquely minimizes the anisotropic perimeter among (1,2)-clusters and the triod among (1,3)-clusters in R^2, with the result extended to general anisotropies by approximation.

References

12 extracted · 12 resolved · 1 Pith anchors

[1] The standard lens cluster inR2 uniquely minimizes relative perimeter 2025 · doi:10.1090/btran/186

[2] Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints 1976

[3] The standard double soap bubble inR2 uniquely minimizes perimeter 1993 · doi:10.2140/pjm.1993.159.47

[4] On the Steiner property for planar minimizing clusters. The anisotropic case 2023 · doi:10.5802/jep.238

[5] Proof of the double bubble conjecture 2002 · doi:10.48550/arxiv.math/0406017

Receipt and verification

First computed	2026-05-20T00:01:40.808088Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

da089d30b8227a621c2aa630523b6fd58475e3bcb135ca3e53908cae2ee7d1a5

Aliases

arxiv: 2604.03510 · arxiv_version: 2604.03510v2 · doi: 10.48550/arxiv.2604.03510 · pith_short_12: 3IEJ2MFYEJ5G · pith_short_16: 3IEJ2MFYEJ5GEHBK · pith_short_8: 3IEJ2MFY

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/3IEJ2MFYEJ5GEHBKUYYFEO3P2W \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: da089d30b8227a621c2aa630523b6fd58475e3bcb135ca3e53908cae2ee7d1a5

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a1378d6902d762eba06905f770d427797ca429fe858d9c7f521370f602a81344",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AP",
    "submitted_at": "2026-04-03T23:12:08Z",
    "title_canon_sha256": "e928e09b6a522a03b87879c8c87843359d372d915b3f5598e6ffeb0c351e2550"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.03510",
    "kind": "arxiv",
    "version": 2
  }
}