arXiv subjects

Ludek Zajicek

Publications and source records attributed to Ludek Zajicek.

14 recordsLinked to original sources

arXiv2026

A characterization of the local structure of two-dimensional sets with positive reach

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 $k\leq d$ 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 $A\subset R^d$ 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 $1\leq k<d$, and $A$ is $k$-dimensional, it can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces.

math.MG↗

arXiv2022

Frechet differentiability via partial Frechet differentiability

Let $X_1, \dots, X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times\dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fr\' echet differentiable but is not Fr\' echet differentiable. Our results imply that if $X_1, \dots, X_{n-1}$ are Asplund spaces and $f$ is continuous (resp. Lipschitz) on $X$, then $A_f$ is a first category set (resp. a $σ$-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f:X \to Y$ is a Lipschitz mapping, then the set of all points $x \in X$ at which $f$ is G\^ ateaux differentiable, is Fr\' echet differentiable along a closed subspace of finite codimension but is not Fr\' echet differentiable, is $σ$-upper porous. A number of related more general results are also proved.

math.FA↗

arXiv2022

On sets of discontinuities of functions continuous on all lines

Answering a question asked by K.C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a set $M \subset \operatorname{graph} f$ nowhere dense in $\operatorname{graph} f$ such that there does not exist any linearly continuous function on $\mathbb R^2$ (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 $\mathbb R^n$ 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. Slobodnik in 1976 is not sufficient. We also prove an analogon of this Slobodnik's result in separable Banach spaces.

math.FA↗

arXiv2021

On the set of points at which an increasing continuous singular function has a nonzero finite derivative

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 $F_σ$ null subset of $[0,1]$.

math.FA↗

arXiv2019

Smallness of the set of critical values of distance functions in two-dimensional Euclidean and Riemannian spaces

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.

math.MG↗

arXiv2016

On the structure of sets with positive reach

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 ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset \neq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces.

math.MG↗

arXiv2013

Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.

math.FA↗

arXiv2012

G\^ ateaux and Hadamard differentiability via directional differentiability

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x \in X$ except a $σ$-directionally porous set: If the one-sided Hadamard directional derivative $f'_{H+}(x,u)$ exists in all directions $u$ from a set $S_x \subset X$ whose linear span is dense in $X$, then $f$ is Hadamard differentiable at $x$. This theorem improves and generalizes a recent result of A.D. Ioffe, in which the linear span of $S_x$ equals $X$ and $Y = \R$. An analogous theorem, in which $f$ is pointwise Lipschitz, and which deals with the usual one-sided derivatives and G\^ ateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which $f$ is supposed to be Lipschitz.

math.FA↗

arXiv2012

Hadamard differentiability via G\^ ateaux differentiability

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ a mapping. We prove that there exists a $σ$-directionally porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is Lipschitz at $x$, and $f$ is Gâteaux differentiable at $x$, then $f$ is Hadamard differentiable at $x$. If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux differentiable at all points, then $f$ is Hadamard differentiable at all points except a set which is $σ$-directionally porous set (and so is Aronszajn null, Haar null and $Γ$-null). Consequently, an everywhere G\^ ateaux differentiable $f: \R^n \to Y$ is Fr\' echet differentiable except a nowhere dense $σ$-porous set.

math.FA↗

arXiv2010

A note on propagation of singularities of semiconcave functions of two variables

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in $\R^n$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $ψ(x) = (x, y_1(x)-y_2(x)), x \in [0,α]$, where $y_1$, $y_2$ are convex and Lipschitz on $[0,α]$. In other words: singularities propagate along arcs with finite turn.

math.CA↗

arXiv2009

Properties of distance functions on convex surfaces and applications

If $X$ is a convex surface in a Euclidean space, then the squared intrinsic distance function $\dist^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function $\dist^2(x,F)$ from a closed set $F \subset X$. Applications concerning $r$-boundaries (distance spheres) and the ambiguous locus (exoskeleton) of a closed subset of a convex surface are given.

math.MG↗

arXiv2009

Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's result is an easy consequence of the only fact that the distance function $d(x)= \dist(x,F)$ is locally DC and has no stationary point in $\R^n\setminus F$. Using this observation, we show that Ferry's (and even Fu's) result extends to sufficiently smooth normed linear spaces $X$ with $\dim X \in \{2,3\}$ (e.g., to $\ell^p_n, n=2,3, p\geq 2$), which improves and generalizes a result of R. Gariepy and W.D. Pepe (1972). By the same method we also generalize Fu's result to Riemannian manifolds and improve a result of K. Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.

math.MG↗

arXiv2008

On extensions of d.c. functions and convex functions

We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems, concerning extendability of continuous convex functions from a closed subspace of a normed linear space, complement recent results of J.Borwein, V.Montesinos and J.Vanderwerff.

math.FA↗

arXiv2008

Curves in Banach spaces which allow a $C^2$ parametrization

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fréchet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.

math.CA↗