We show that for every compact domain in a Euclidean space with d.c. (delta-convex) boundary there exists a unique Legendrian cycle such that the associated curvature measures fulfil a local version of the Gauss-Bonnet formula. This was known in dimensions two and three and was open in higher dimensions. In fact, we show this property for a larger class of sets including also lower-dimensional sets. We also describe the local index function of the Legendrian cycles and we show that the associated curvature measures fulfill the Crofton formula.Comment: 22 pp, corrected versio

Abstract. The class WDC(M ) consists of all subsets of a smooth manifold M that may be expressed in local coordinates as certain sublevel sets of DC (differences of convex) functions. If M is Riemanian and G is a group of isometries acting transitively on the sphere bundle SM , we define the invariant curvature measures of compact WDC subsets of M , and show that pairs of such subsets are subject to the array of kinematic formulas known to apply to smoother sets. Restricting to the case (M, G) = (R n , SO(n)), this extends and subsumes Federer's theory of sets with positive reach in an essential way. The key technical point is equivalent to a sharpening of a classical theorem of Ewald, Larman, and Rogers characterizing the dimension of the set of directions of line segments lying in the boundary of a given convex body.

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...

We study the reflexivity, the uniform convexity, the Daugavet property and the Radon-Nikodym property of the generalized Lebesgue spaces L p(x) .

In the present article we provide a sufficient condition for a closed set F ∈ R d to have the following property which we call c-removability: Whenever a function f : R d → R is locally convex on the complement of F , it is convex on the whole R d . We prove that no generalized rectangle of positive Lebesgue measure in R 2 is c-removable.Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor [J. Math. Anal. Appl. 365 (2010)]: Assume the closed set F ⊂ R d is such that any locally convex function defined on R d \ F has a unique convex extension on R d . Is F necessarily intervally thin (a notion of smallness of sets defined by their "essential transparency" in every direction)?We prove the answer is negative by finding a counterexample in R 2 .2010 Mathematics Subject Classification. 26B25, 52A20.