On the characterization of some classes of proximally smooth sets

Crasta, Graziano; Fragalà, Ilaria Fragalà

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 710-727.

We provide a complete characterization of closed sets with empty interior and positive reach in $R^2$. As a consequence, we characterize open bounded domains in $R^2$ whose high ridge and cut locus agree, and hence $C^1$ planar domains whose normal distance to the cut locus is constant along the boundary. The latter result extends to convex domains in $R^n$.

Classification : 26B25, 26B05, 53A05
Mots clés : Distance function, proximal smoothness, positive reach, cut locus, central set, skeleton, medial axis

P. Albano, P. Cannarsa, Khai T. Nguyen and C. Sinestrari, Singular gradient flow of the distance function and homotopy equivalence. Math. Ann. 356 (2013) 23–43. | DOI | MR | Zbl

G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differ. Equ. 2 (1994) 17–27. | DOI | MR | Zbl

A.D. Alexandrov, Uniqueness theorems for surfaces in the large. I, II. Amer. Math. Soc. Transl. 31 (1962) 341–388. | Zbl

L. Ambrosio, Geometric Evolution Problems, Distance Function and Viscosity Solutions. Calculus of Variations and Partial Differential Equations (Pisa, 1996), Springer, Berlin (2000) 5–93. | MR | Zbl

V. Bangert, Sets with positive reach. Arch. Math. (Basel) 38 (1982) 54–57. | DOI | MR | Zbl

G. Bellettini, M. Masala and M. Novaga, A conjecture of De Giorgi on the square distance function. J. Convex Anal. 14 (2007) 353–359. | MR | Zbl

F. Bernard, L. Thibault and N. Zlateva, Characterizations of prox-regular sets in uniformly convex Banach spaces. J. Convex Anal. 13 (2006) 525–559. | MR | Zbl

C.J. Bishop, Tree-like decompositions of simply connected domains. Rev. Mat. Iberoam. 28 (2012) 179–200. | DOI | MR | Zbl

C.J. Bishop and H. Hakobyan, A central set of dimension 2. Proc. Amer. Math. Soc. 136 (2008) 2453–2461. | DOI | MR | Zbl

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function. Calc. Var. Partial Differ. Equ. 3 (1995) 273–298. | DOI | MR | Zbl

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton−Jacobi Equations and Optimal Control. Vol. 58 of Progr. Nonlin. Differ. Equ. Appl. Birkhäuser, Boston (2004). | MR | Zbl

P. Cannarsa, P. Cardaliaguet and E. Giorgieri, Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles. Trans. Amer. Math. Soc. 359 (2007) 2741–2775 (electronic). | DOI | MR | Zbl

A. Cellina, Minimizing a functional depending on $\nabla u$ and on $u$. Ann. Inst. Henri Poincaré, Anal. Non Lin. 14 (1997) 339–352. | DOI | Numdam | MR | Zbl

H.I. Choi, S.W. Choi and H.P. Moon, Mathematical theory of medial axis transform. Pacific J. Math. 181 (1997) 57–88. MR 1491036 | DOI | MR | Zbl

F.H. Clarke, Optimization and Nonsmooth Analysis. Canadian Math. Soc. Ser. Math. John Wiley and Sons, Toronto (1983). | MR

F.H. Clarke, R.J. Stern and P.R. Wolenski, Proximal smoothness and the lower-$C^2$ property. J. Convex Anal. 2 (1995) 117–144. | MR | Zbl

A. Colesanti and D. Hug, Steiner type formulae and weighted measures of singularities for semi-convex functions. Trans. Amer. Math. Soc. 352 (2000) 3239–3263 (electronic). | DOI | MR | Zbl

G. Colombo and K.T. Nguyen, On the structure of the minimum time function. SIAM J. Control Optim. 48 (2010) 4776–4814. | DOI | MR | Zbl

G. Colombo and L. Thibault, Prox-regular Sets and Applications. Handbook of Nonconvex Analysis and Applications. Int. Press, Somerville, MA (2010) 99–182. | MR | Zbl

G. Colombo, A. Marigonda and P.R. Wolenski, Some new regularity properties for the minimal time function. SIAM J. Control Optim. 44 (2006) 2285–2299. | DOI | MR | Zbl

G. Crasta and I. Fragalà, A new symmetry criterion based on the distance function and applications to PDE’s. J. Differ. Equ. 255 (2013) 2082–2099. | DOI | MR | Zbl

G. Crasta and I. Fragalà, A symmetry problem for the infinity Laplacian. To appear in Int. Math. Res. Not. (2014). Doi: | DOI | MR

G. Crasta and I. Fragalà, On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results. Arch. Ration. Mech. Anal. 218 (2015) 1577–1607. | arXiv | DOI | MR | Zbl

G. Crasta and A. Malusa, The distance function from the boundary in a Minkowski space. Trans. Amer. Math. Soc. 359 (2007) 5725–5759. | DOI | MR | Zbl

G. Crasta, I. Fragalà and F. Gazzola, A sharp upper bound for the torsional rigidity of rods by means of web functions. Arch. Ration. Mech. Anal. 164 (2002) 189–211. | DOI | MR | Zbl

G. Crasta, I. Fragalà and F. Gazzola, On the role of energy convexity in the web function approximation. Nonlin. Differ. Equ. Appl. 12 (2005) 93–109. | DOI | MR | Zbl

G. Crasta, I. Fragalà and F. Gazzola, Some estimates of the torsional rigidity of heterogeneous rods. Math. Nach. 280 (2007), 242–255. | DOI | MR | Zbl

W.L.F. Degen, Exploiting curvatures to compute the medial axis for domains with smooth boundary. Comput. Aid. Geom. Design 21 (2004) 641–660. | DOI | MR | Zbl

M. Delfour and J.-P. Zolésio, Shape analysis via oriented distance functions. J. Functional Anal. 123 (1994) 129–201. | DOI | MR | Zbl

H. Federer, Curvature measures. Trans. Amer. Math. Soc. 93 (1959) 418–491. | DOI | MR | Zbl

I. Fragalà, F. Gazzola and B. Kawohl, Overdetermined problems with possibly degenerate ellipticity, a geometric approach. Math. Z. 254 (2006) 117–132. | DOI | MR | Zbl

D.H. Fremlin, Skeletons and central sets. Proc. London Math. Soc. 74 (1997) 701–720. | DOI | MR | Zbl

J. Frerking and U. Westphal, On a property of metric projections onto closed subsets of Hilbert spaces. Proc. Amer. Math. Soc. 105 (1989) 644–651. | DOI | MR | Zbl

J.H.G. Fu, Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52 (1985) 1025–1046. | MR | Zbl

E. Giusti, Metodi diretti nel Calcolo delle Variazioni. Unione Matematica Italiana, Bologna (1994). | MR | Zbl

D. Hug, Generalized curvature measures and singularities of sets with positive reach. Forum Math. 10 (1998) 699–728. | MR | Zbl

D. Hug, G. Last and W. Weil, A local Steiner-type formula for general closed sets and applications. Math. Z. 246 (2004) 237–272. | DOI | MR | Zbl

J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus. Trans. Amer. Math. Soc. 353 (2001) 21–40. | DOI | MR | Zbl

Y.Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton–Jacobi equations. Commun. Pure Appl. Math. 58 (2005) 85–146. | DOI | MR | Zbl

A. Lieutier, Any Open Bounded Subset of ${𝐑}^n$ Has The same Homotopy Type as its Medial Axis. Proc. of 8th ACM Sympos. Solid Modeling Appl. ACM Press (2003) 65–75.

C. Mantegazza and A.C. Mennucci, Hamilton−Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47 (2003) 1–25. | DOI | MR | Zbl

C. Nour, R.J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition. J. Convex Anal. 16 (2009) 501–514. | MR | Zbl

J. Rataj and M. Zähle, Mixed curvature measures for sets of positive reach and a translative integral formula. Geom. Dedicata 57 (1995) 259–283. | DOI | MR | Zbl

R.T. Rockafellar, Favorable Classes of Lipschitz-Continuous Functions in Subgradient Optimization. Progr. Nondiffer. Optimiz. IIASA Collaborative Proc. Ser. CP-82, vol. 8 Int. Inst. Appl. Systems Anal., Laxenburg (1982) 125–143. | MR | Zbl

J. Serrin, A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971) 304–318. | DOI | MR | Zbl

C. Thäle, 50 years sets with positive reach − a survey. Surv. Math. Appl. 3 (2008) 123–165. | MR | Zbl

F. Wolter, Cut locus and medial axis in global shape interrogation and representation. Tech. Report Memorandum 92-2, MIT, Department of Ocean Engineering, Design Laboratory, Cambridge, MA (1993).

M. Zähle, Integral and current representation of Federer’s curvature measures. Arch. Math. (Basel) 46 (1986) 557–567. | DOI | MR | Zbl