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We call $$I_{\\rho^{2}}:=\\inf_{B_{\\rho}}I(u) \\ $$ where $B_{\\rho}=\\{u\\in H^{m}(\\R^{N}):\\|u\\|_{2}=\\rho\\},$ and $I(u)=1/2\\|u\\|^{2}_{D^{m,2}}+T(u)$, $T$ fulfilling general assumptions. We show that the regularity of the function $$(0,\\infty)\\ni s \\mapsto I_{s^2}\\,$$ and the behaviour of $\\frac{I_{s^2}}{s^2}$ in the neighborhood of zero allows to prove the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to tra"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1007.4139","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-07-23T14:30:50Z","cross_cats_sorted":[],"title_canon_sha256":"4c17f71925dd8b085bf7e5e8b1ac0186ca091fa3100890849c68bb489b0a2e6b","abstract_canon_sha256":"81ef998f1404f871508b9da11d134f7e7b6bbceb2bd3820c45401f2a8920169b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:35:14.655126Z","signature_b64":"xquQZd40BKP9w50IMHYcygWtB+kIyA9+Y68hm+325F9ogWvQl/twSXSegX31WYMT9FAjsGQ7focr+Zn3c5XBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0dbe470db66b367f15ea6790b239dd9c147c34cade5851b1f20bbca95c69f767","last_reissued_at":"2026-05-18T04:35:14.654454Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:35:14.654454Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On translation invariant constrained minimization problems with application to Schr\\\"odinger-Poisson equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gaetano Siciliano, Jacopo Bellazzini","submitted_at":"2010-07-23T14:30:50Z","abstract_excerpt":"In this paper we study the existence of minimizers for a class of constrained minimization problems that are invariant under translations. We call $$I_{\\rho^{2}}:=\\inf_{B_{\\rho}}I(u) \\ $$ where $B_{\\rho}=\\{u\\in H^{m}(\\R^{N}):\\|u\\|_{2}=\\rho\\},$ and $I(u)=1/2\\|u\\|^{2}_{D^{m,2}}+T(u)$, $T$ fulfilling general assumptions. We show that the regularity of the function $$(0,\\infty)\\ni s \\mapsto I_{s^2}\\,$$ and the behaviour of $\\frac{I_{s^2}}{s^2}$ in the neighborhood of zero allows to prove the strong subadditivity inequality. 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