Abstract
The metric projection to a set \(\emptyset \neq X\subset \mathbb {R}^d\) is not determined everywhere unless X is convex and closed. Sets with positive reach are sets with the property that the metric projection is defined on some open neighbourhood of X.
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References
V. Bangert, Sets with positive reach. Arch. Math. (Basel) 38, 54–57 (1982)
H. Federer, Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
J.H.G. Fu, Tubular neighborhoods in Euclidean spaces. Duke Math. J. 52, 1025–1046 (1985)
J.H.G. Fu, Curvature measures and generalized Morse theory. J. Differ. Geom. 30, 619–642 (1989)
J.H.G. Fu, Curvature measures for subanalytic sets. Am. J. Math. 116, 819–880 (1994)
D. Hug, G. Last, W. Weil, A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272 (2004)
N. Kleinjohann, Nächste Punkte in der Riemanschen Geometrie. Math. Z. 176, 327–344 (1981)
A. Lytchak, On the geometry of subsets of positive reach. Manuscripta Math. 115, 199–205 (2004)
A. Lytchak, Almost convex subsets. Geom. Dedicata 115, 201–218 (2005)
G. Matheron, Random Sets and Integral Geometry (Wiley, New York, 1975)
J. Rataj, L. Zajíček, On the structure of sets with positive reach. Math. Nachr. 290, 1806–1829 (2017)
R. Sulanke, P. Wintgen, Differentialgeometrie und Faserbündel (Birkhäuser Verlag, Basel/Stuttgart, 1972)
H. Weyl, On the volume of tubes. Am. J. Math. 1939, 461–472 (1939)
M. Zähle, Curvature measures and random sets II. Probab. Th. Rel. Fields 71, 37–58 (1986)
M. Zähle, Integral and current representation of Federer’s curvature measures. Arch. Math. 46, 557–567 (1986)
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Rataj, J., Zähle, M. (2019). Sets with Positive Reach. In: Curvature Measures of Singular Sets. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-18183-3_4
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DOI: https://doi.org/10.1007/978-3-030-18183-3_4
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