On the structure of WDC sets

Abstract

WDC sets in ℝ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set of topological dimension ≤ can be decomposed into the union of two sets, one of them being a -dimensional DC manifold open in , and the other can be covered by finitely many DC surfaces of dimension − 1. We also characterize locally WDC sets among closed Lipschitz domains and among lowerdimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane. K E Y W O R D S DC aura, DC domain, DC manifold, deformation retraction, Gauss-Bonnet formula, Lipschitz manifold, WDC set M S C ( 2 0 1 0 ) 26B25, 53C65 INTRODUCTIONFederer in his fundamental paper [10] unified the approaches of convex and differential geometry, introducing curvature measures for sets with positive reach and proving the kinematic formulas. Quite recently, curvature measures have been defined for a substantially larger class of so-called (locally) WDC sets [23], and the corresponding kinematic formulas have been proved [16]. The basic difference between the two named set classes is that, while sets with positive reach are closely related to semiconvex functions of several variables, WDC sets are related to DC functions (i.e., differences of two convex functions) instead. Following [14], we say that a locally Lipschitz function ∶ ℝ → [0, ∞) is an aura for a set ⊂ ℝ if = −1 {0} and 0 is a weakly regular value of (i.e., there exist no sequences → and → 0 such that ( ) > 0 = ( ) and ∈ ( ) are generalized gradients in the Clarke sense).This notion is motivated by the fact that has locally positive reach if and only if has a semiconvex aura [1]. By the definition, is WDC if and only if it has a DC aura. So each set with locally positive reach is a WDC set.Because of the theory built in [16,23], the following rough question naturally arises: What is the structure of a general WDC set? Note that, in contrast with sets with positive reach which are defined by the geometrically illustrative "unique footpoint" property, there seems to be no purely geometric property characterizing WDC sets. Also, there is a number of results on the structure of sets of positive reach, see, e.g., [10,21], or the recent article [24]. In the present article we prove some results on WDC sets, which are analogous to these results on sets of positive reach. 1596POKORNÝ ET AL. Boundaries of WDC setsWe show that the boundary of a compact WDC set in ℝ can be covered by finitely many DC hypersurfaces (i.e., graphs of Lipschitz DC functions of − 1 variables, see Proposition 6.1). Also, we show that a closed Lipschitz domain is locally WDC...

On sets in Rd with DC distance function

“…However, using Proposition 4.2, we easily obtain: Proof. By our assumptions, f := dist (·, F ) is a DC function on R d and f (x) = 0 for every x ∈ C. So, by [12,Crollary 5.4] it is sufficient to prove that for each x ∈ C there exists y * ∈ ∂f (x) with |y * | > ε := 1/4, where ∂f (x) is the Clarke generalized gradient of f at x (see [3, p. 27]). To this end, suppose to the contrary that x ∈ C and ∂f (x) ⊂ B(0, 1/4).…”

“…Motivated by a natural question, for which non DC functions g (4.8) holds, we present the following result, whose proof is implicitly contained in the proof of [12,Proposition 6.6]; see Remark 4.4 below. Remark 4.4.…”

“…Remark 4.4. One implication of [12,Proposition 6.6] gives that if A is as in Proposition 4.3 (or, more generally, A is a Lipschitz manifold of dimension 0 < k < d; see [12,Definition 2.4] for this notion) and A is WDC, then g is DC (or is a DC manifold of dimension 0 < k < d, respectively). The proof of this implication works with an aura f = f M of a set M , but under the assumption that A ∈ D d , the proof clearly also works, if we use the distance function d A instead of f .…”

Curvatures for unions of WDC sets

“…Suppose that 𝑃 is a DC graph in ℝ 2 with 𝐾 = 𝑣 ⊥ . Then the following is true (see [14,Remark 7.1] for the proof, note that Lipschitzness of 𝑓 is not needed in the proof): for…”

“…We will also use the following easy fact which we state without a proof. Let us recall some definitions and notation from [14,Definition 7.4]. If 𝑧 ∈ ℝ 2 and 𝑣 ∈ 𝕊 1 , we denote by 𝛾 𝑧,𝑣 the unique orientation-preserving isometry on ℝ 2 that maps 0 to 𝑧 and (1,0) to 𝑧 + 𝑣.…”