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Anisotropic tubular neighborhoods of sets

Chambolle, Antonin; Lussardi, Luca; Villa, Elena (2021), Anisotropic tubular neighborhoods of sets, Mathematische Zeitschrift, 299, p. 1257–1274. 10.1007/s00209-021-02715-9

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
Mathematische Zeitschrift
Volume
299
Publisher
Springer
Pages
1257–1274
Publication identifier
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Author(s)
Chambolle, Antonin cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Centre de Mathématiques Appliquées - Ecole Polytechnique [CMAP]
Lussardi, Luca
Dipartimento di Scienze Matematiche
Villa, Elena
Dipartimento di Matematica "Federico Enriques"
Abstract (EN)
Let E ⊂ R^N be a compact set and C ⊂ R^N be a convex body with 0 ∈ int C. We prove that the topological boundary of the anisotropic enlargement E + rC is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function V_E(r) := |E + rC| proving a formula for the right and the left derivatives at any r > 0 which implies that V E is of class C^1 up to a countable set completely characterized. Moreover, some properties on the second derivative of V_E are proved.
Subjects / Keywords
Rectifiability; anisotropic outer Minkowski content; viscosity solutions