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Keywords:
distance function; WDC set; DC function; DC aura; Borel complexity
distance function; WDC set; DC function; DC aura; Borel complexity
Summary:
We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset {\mathbb R}^2$. We prove that, for such $A$, the distance function $d_A= {\rm dist}(\cdot,A)$ is a ``DC aura'' for $A$, which implies that each closed locally WDC set in ${\mathbb R}^2$ is a WDC set. Another consequence is that compact WDC subsets of ${\mathbb R}^2$ form a Borel subset of the space of all compact sets.
References:
[1] Bangert V.: Sets with positive reach. Arch. Math. (Basel) 38 (1982), no. 1, 54–57. DOI 10.1007/BF01304757 | MR 0646321
[2] Cannarsa P., Sinestrari C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, 58, Birkhäuser, Boston, 2004. MR 2041617
[3] Clarke F. H.: Optimization and Nonsmooth Analysis. Classics in Applied Mathematics, 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1990. MR 1058436 | Zbl 0696.49002
[4] DeVore R. A., Lorentz G. G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften, 303, Springer, Berlin, 1993. DOI 10.1007/978-3-662-02888-9_3 | MR 1261635
[5] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[6] Fu J. H. G.: Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52 (1985), no. 4, 1025–1046. MR 0816398 | Zbl 0592.52002
[7] Fu J. H. G.: Integral geometric regularity. in Tensor Valuations and Their Applications in Stochastic Geometry and Imaging, Lecture Notes in Math., 2177, Springer, Cham, 2017, pages 261–299. MR 3702376
[8] Fu J. H. G., Pokorný D., Rataj J.: Kinematic formulas for sets defined by differences of convex functions. Adv. Math. 311 (2017), 796–832. DOI 10.1016/j.aim.2017.03.003 | MR 3628231
[9] Hartman P.: On functions representable as a difference of convex functions. Pacific J. Math. 9 (1959), 707–713. DOI 10.2140/pjm.1959.9.707 | MR 0110773
[10] Pokorný D., Rataj J.: Normal cycles and curvature measures of sets with d.c. boundary. Adv. Math. 248 (2013), 963–985. DOI 10.1016/j.aim.2013.08.022 | MR 3107534
[11] Pokorný D., Rataj J., Zajíček L.: On the structure of WDC sets. Math. Nachr. 292 (2019), no. 7, 1595–1626. DOI 10.1002/mana.201700253 | MR 3982330
[12] Pokorný D., Zajíček L.: On sets in ${\mathbb R}^d$ with DC distance function. J. Math. Anal. Appl. 482 (2020), no. 1, 123536, 14 pages. DOI 10.1016/j.jmaa.2019.123536 | MR 4015277
[13] Srivastava S. M.: A Course on Borel Sets. Graduate Texts in Mathematics, 180, Springer, New York, 1998. DOI 10.1007/978-3-642-85473-6 | MR 1619545 | Zbl 0903.28001
[14] Tuy H.: Convex Analysis and Global Optimization. Springer Optimization and Its Applications, 110, Springer, Cham, 2016. DOI 10.1007/978-3-319-31484-6 | MR 3560830 | Zbl 0904.90156
[15] Veselý L., Zajíček L.: Delta-convex mappings between Banach spaces and applications. Dissertationes Math. (Rozprawy Mat.) 289 (1989), 52 pages. MR 1016045
[16] Zähle M.: Curvature measures and random sets. II. Probab. Theory Relat. Fields 71 (1986), no. 1, 37–58. DOI 10.1007/BF00366271 | MR 0814660
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