Tubular neighborhoods in Euclidean spaces

Tube Formulas and Complex Dimensions of Self-Similar Tilings

“…with g = 6 , D = log 3 2, and p = 2π/ log 3. It is this form of the tube formula which is closer in appearance to the following examples.…”

“…See [1, §6.6-6.9] and the book [9]. Federer's tube formula has since been extended in various directions by a number of researchers in integral geometry and geometric measure theory, including [36], [37], [42], [43], [6], [7], [38], and most recently (and most generally) in [13]. The books [9] and [37] contain extensive endnotes with further information and many other references.…”

A Morse-Sard theorem for the distance function on Riemannian manifolds

“…Remark 1.1. Twenty years ago, Fu proved in [6] that the set of all critical values of the distance function from a compact subset in R 3 is of Lebesgue measure zero. Following Whitney [17], S. Ferry [5] gave a counterexample to this result in dimension greater than three.…”

Sets of constant distance from a Jordan curve

“…In [3], Ferry showed, among other results, that ∂ (A) is a 1-manifold for almost all . Fu [4] generalized Ferry's results, and proved that for all outside a compact set of zero 1/2-dimensional Hausdorff measure, ∂ (A) is a Lipschitz 1-manifold. Papers [3] and [4] include theorems in higher dimensional Euclidean spaces; the work for dimensions n ≥ 3 is more demanding.…”