The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in R^d. We also present a more elementary proof of a recent result of A. Lytchak which describes for k≤ d the local structure of k-dimensional sets with positive reach A in R^d at points where the tangent cone of A is k-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in R^d. Our method also shows that, for any set A⊂ R^d with positive reach, the set of points at which the tangent cone of A is k-dimensional is locally contained in a k-dimensional C^1,1 surface. As a consequence we obtain that if 1≤ k<d, and A is k-dimensional, it can be covered by countably many k-dimensional C^1,1 surfaces.

Our note is a complement to recent articles [17] (2011) and [18] (2013) by M. Jim & eacute;nez-Sevilla and L. Sanchez-Gonzalez which generalise (the basic statement of) the classical Whitney extension theorem for C-1-smooth real functions on Rn to the case of real functions on X ([17]) and to the case of mappings from X to Y ([18]) for some Banach spaces X and Y. Since the proof from [18] contains a serious flaw, we supply a different more transparent detailed proof under (probably) slightly stronger assumptions on X and Y. Our proof gives also extensions results from special sets (e.g. Lipschitz submanifolds or closed convex bodies) under substantially weaker assumptions on X and Y. Further, we observe that the mapping F is an element of C-1(X;Y) which extends f given on a closed set A subset of X can be, in some cases, C-infinity-smooth (or C-k-smooth with k > 1) on X\A. Of course, also this improved result is weaker than Whitney's result (for X = R-n, Y = R) which asserts that F is even analytic on X\A. Further, following another Whitney's article and using the above results, we prove results on extensions of C-1-smooth mappings from open ("weakly") quasiconvex subsets of X. Following the above mentioned articles [17], [18] we also consider the question concerning the Lipschitz constant of F if f is a Lipschitz mapping. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We prove that, in any separable uniformly smooth Banach space, the boundary of every a-convex set (in the Efimov-Stechkin sense) is Gamma-null (in the Lindenstrauss-Preiss sense). Using well-known results, we obtain that the same is true for proximally smooth sets (in the sense of Clarke, Stern and Wolenski) and for uniformly prox-regular sets (in the sense of Poliquin, Rockafellar and Thibault) in any separable Banach space which is both uniformly convex and uniformly smooth.

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021, [2]). We show how older results on semiconvex functions with modulus omega easily imply extension theorems for C1,omega-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of C1,omega-smooth functions from open quasiconvex sets. They proved also an extension theorem for C1,+ B-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for C1,+ and C1,+ loc-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article. (c) 2023 Elsevier Inc. All rights reserved.

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021, [2]). We show how older results on semiconvex functions with modulus ω easily imply extension theorems for C1,ω-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of C1,ω-smooth functions from open quasiconvex sets. They proved also an extension theorem for CB1,+-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for C1,+ and Cloc1,+-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.

Our paper is a complement to a recent article by D. Azagra and C. Mudarra (2021). We show how older results on semiconvex functions with modulus ω easily imply extension theorems for C^1,ω-smooth functions on super-reflexive Banach spaces which are versions of some theorems of Azagra and Mudarra. We present also some new interesting consequences which are not mentioned in their article, in particular extensions of C^1,ω-smooth functions from open quasiconvex sets. They proved also an extension theorem for C_B^1,+-smooth functions (i.e., functions with uniformly continuous derivative on each bounded set) on Hilbert spaces. Our version of this theorem and new extension results for C^1,+ and C_loc^1,+-smooth functions (i.e., functions with uniformly, resp. locally uniformly continuous derivative), all of which are proved on arbitrary super-reflexive Banach spaces, are further main contributions of our paper. Some of our proofs use main ideas of the article by D. Azagra and C. Mudarra, but all are formally completely independent on their article.

Let X-1 , ... , X-n be Banach spaces and f a real function on X = X-1 x x X-n. Let A(f) be the set of all points x is an element of X at which f is partially Frechet differentiable but is not Frechet differentiable. Our results imply that if X-1 , ... , Xn-1 are Asplund spaces and f is continuous (respectively Lipschitz) on X, then A(f) is a first category set (respectively a sigma-upper porous set). We also prove that if X, Y are separable Banach spaces and f : X -> Y is a Lipschitz mapping, then there exists a sigma-upper porous set A subset of X such that f is Frechet differentiable at every point x is an element of X \ A at which it is Frechet differentiable along a closed subspace of finite codimension and G & aacute;teaux differentiable. A number of related more general results are also proved.

Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\sigma}$ null subset of $[0,1]$.

Answering a question asked by K. C. Ciesielski and T. Glatzer in 2013, we construct a C1-smooth function f on [0,1] and a closed set M subset of graphf nowhere dense in graphf such that there does not exist any linearly continuous function on R2 (i.e., function continuous on all lines) which is discontinuous at each point of M. We substantially use a recent full characterization of sets of discontinuity points of linearly continuous functions on Rn proved by T. Banakh and O. Maslyuchenko in 2020. As an easy consequence of our result, we prove that the necessary condition for such sets of discontinuities proved by S. G. Slo-bodnik in 1976 is not sufficient. We also prove an analogue of this Slobodnik's result in separable Banach spaces.

We give a complete characterization of closed sets F ⊂ ℝ 2 whose distance function d F := dist(·, F ) is DC (i.e., is the difference of two convex functions on ℝ 2 ). Using this characterization, a number of properties of such sets is proved.

Let G ⊂ℝ^n be an open convex set which is either bounded or contains a translation of a convex cone with nonempty interior. It is known that then, for every modulus ω, every function on G which is both semiconvex and semiconcave with modulus ω is (globally) C^1,ω-smooth. We show that this result is optimal in the sense that the assumption on G cannot be relaxed. We also present direct short proofs of the above mentioned result and of some its quantitative versions. Our results have immediate consequences concerning (i) a first-order quantitative converse Taylor theorem and (ii) the problem whether f∈ C^1,ω(G) whenever f is continuous and smooth in a corresponding sense on all lines. We hope that these consequences are of an independent interest.

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset \mathbb{R}^2$. We prove that, for such $A$, the distance function $d_A= {\rm dist}(\cdot,A)$ is a `DC aura' for $A$, which implies that each locally WDC set in $\mathbb{R}^2$ is a WDC set. An another consequence is that compact WDC subsets of $\mathbb{R}^2$ form a Borel subset of the space of all compact sets.

We study closed sets F⊂Rd whose distance function dF≔dist(⋅,F) is DC (i.e., is the difference of two convex functions on Rd). Our main result asserts that if F⊂R2 is a graph of a DC function g:R→R, then F has the above property. If d>1, the same holds if g:Rd−1→R is semiconcave, however the case of a general DC function g remains open.

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

We strengthen and generalize results of J. M. Borwein [Partially monotone operators and the generic differentiability of convex-concave and biconvex mappings, Israel J. Math. 54 (1986) 42-50] and of A. Ioffe and R. E. Lucchetti [Typical convex program is very well posed, Math. Program. 104 (2005) 483-499] on Frechet and Gateaux differentiability of saddle and biconvex functions (and operators). For example, we prove that in many cases (also in some cases which were not considered before) these functions (and operators) are Frechet differentiable except for a G-null, s-lower porous set. Moreover, we prove these results for more general "partially convex (up or down)" functions and operators defined on the product of n Banach spaces.

WDC sets in ${\mathbb R}^d$ 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 $M$ of topological dimension $k\leq d$ can be decomposed into the union of two sets, one of them being a $k$-dimensional DC manifold open in $M$, and the other can be covered by finitely many DC surfaces of dimension $k-1$. We also characterize locally WDC sets among closed Lipschitz domains and among lower-dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.

We prove that each linearly continuous function f on R n (i.e., each function continuous on all lines) belongs to the first Baire class, which answers a problem formulated by K. C. Ciesielski and D. Miller (2016).The same result holds also for f on an arbitrary Banach space X, if f has moreover the Baire property.We also prove (extending a known finite-dimensional result) that such f on a separable X is continuous at all points outside a first category set which is also null in any usual sense.

Let f be a continuous real function on a convex subset of a Banach space. We study what can be said about the semiconcavity (with a general modulus) of f, if we know that the estimate. Delta(2)(h)(f; x) <= omega(parallel to h parallel to) holds, where. Delta(2)(h)(f; x) = f (x + 2h) - 2f (x + h) + f (x) and omega : [0,infinity) -> [0; infinity) is a nondecreasing function right continuous at 0 with omega(0) = 0. A partial answer to this question was given by P. Cannarsa and C. Sinestrari (2004); we prove versions of their result, which are in a sense best possible. We essentially use methods of A. Marchaud, S. B. Stechkin and others, whose results clarify when the inequality vertical bar Delta(2)(h)(f; x)vertical bar <=omega (parallel to h parallel to) implies that f is a C-1 function (and f ' is uniformly continuous with a corresponding modulus of continuity).

A problem asked by the authors in 1989 concerns the natural question, whether one can deduce that a continuous function f on an open convex set D⊂Rn is DC (i.e., is a difference of two convex functions) from the behavior of f “along some special curves φ”. I.M. Prudnikov published in 2014 a theorem (working with convex curves φ in the plane), which would give a positive answer in R2 to our problem. However, in the present note we construct an example showing that this theorem is not correct, and thus our problem remains open in each Rn, n>1.

We give a complete characterization of compact sets with positive reach (proximally C 1 sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “ k ‐dimensional regular part” which is a k ‐dimensional “uniform” C 1, 1 manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.