{"paper":{"title":"Anisotropic isoperimetric double tilings of the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.AP","authors_text":"Emanuele Paolini, Francesco Nobili, Matteo Novaga","submitted_at":"2026-07-12T08:09:11Z","abstract_excerpt":"We study periodic partitions of the plane into two distinct cells minimizing the anisotropic $\\ell_1$-perimeter. For a rectangular lattice $G$, we compute explicitly the $(G,\\ell_1)$-isoperimetric profile and classify all minimizers. When one cell has small area, the optimal tiling is generated by a square and a chipped rectangle, while in the remaining regime the tiling is generated by two adjacent rectangles. Further minimizing over all possible planar lattices, we show that the $\\ell_1$-isoperimetric profile is attained by the Pythagorean double tiling of two axis-aligned squares sharing a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10636","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.10636/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}