Generalized curvature measures and singularities of sets with positive reach

1998 ◽  
Vol 10 (6) ◽  
Author(s):  
Daniel Hug
Keyword(s):  
Positive Reach ◽  
Curvature Measures ◽  
Sets With Positive Reach

scholarly journals On the structure of sets with positive reach

2017 ◽  
Vol 290 (11-12) ◽  
pp. 1806-1829 ◽  
Author(s):  
Jan Rataj ◽  
Luděk Zajíček
Keyword(s):  
Positive Reach ◽  
Sets With Positive Reach

Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (2) ◽  
pp. 341-341 ◽  
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝd. The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝd, where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (02) ◽  
pp. 341
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝ d . The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝ d , where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz ) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

Sets with positive reach

1982 ◽  
Vol 38 (1) ◽  
pp. 54-57 ◽  
Author(s):  
Victor Bangert
Keyword(s):  
Positive Reach ◽  
Sets With Positive Reach
Sign in / Sign up

Export Citation Format

Share Document