L. Zajícek

activity

20062026

most citedOn the structure of sets with positive reach

25 citations · 46 across the 18 of their papers we have counts for

collaborators

21 papers

math.FA2026

The Fréchet derivative is in the first Baire class

Eva Kopecká, Luděk Zaj\'ıček

Let $X$ be a normed linear space, $Y$ a Banach space, $G \subset X$ an open set and $f : G \to Y$ a mapping. We show that the Fréchet derivative $f^{'}$ of $f$ is of the first Baire clas…

math.MG2026

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

Jan Rataj, Ludek Zajicek

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 o…

math.FA2024

On $C^{1}$ Whitney extension theorem in Banach spaces

Michal Johanis, Luděk Zajíček

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jiménez-Sevilla and L. Sánchez-González which generalise (the basic statement of) the clas…

math.FA2022

Frechet differentiability via partial Frechet differentiability

Ludek Zajicek

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…

math.FA2022

On sets of discontinuities of functions continuous on all lines

Ludek Zajicek

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 $M \subset graph f$ nowhere dense…

math.FA2021

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

Marta Kossaczka, Ludek Zajicek

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 no…

math.CA2020

A characterization of sets in $R^{2}$ with DC distance function

Dušan Pokorný, Luděk Zajíček

We give a complete characterization of closed sets $F \subset R^{2}$ whose distance function $d_{F} := dist (\cdot, F)$ is DC (i.e., is the difference of two convex func…

math.CA2019

On sets in $R^{d}$ with DC distance function

Dušan Pokorný, Luděk Zajíček

We study closed sets $F \subset R^{d}$ whose distance function $d_{F} := dist (\cdot, F)$ is DC (i.e., is the difference of two convex functions on $R^{d}$ ). Our…

math.MG2019★ 1 cited

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

Jan Rataj, Ludek Zajicek

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 manif…

math.MG2017

On the structure of WDC sets

Dušan Pokorný, Jan Rataj, Luděk Zajíček

WDC sets in $R^{d}$ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial g…

math.MG2016★ 25 cited

On the structure of sets with positive reach

Jan Rataj, Ludek Zajicek

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 $R^{d}$ . Fu…

math.FA2013

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

Ludek Zajicek

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 functio…

math.FA2012★ 1 cited

G\^ ateaux and Hadamard differentiability via directional differentiability

Ludek Zajicek

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 $σ$ -directional…

math.FA2012

Hadamard differentiability via G\^ ateaux differentiability

Ludek Zajicek

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 \set…

math.CA2010

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

Ludek Zajicek

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…

math.MG2009★ 8 cited

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

Jan Rataj, Ludek Zajicek

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^…

math.MG2009★ 1 cited

Properties of distance functions on convex surfaces and applications

Jan Rataj, Ludek Zajicek

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…

math.FA2008★ 8 cited

On extensions of d.c. functions and convex functions

Libor Vesely, Ludek Zajicek

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 natura…

math.FA2007

Quotients of continuous convex functions on nonreflexive Banach spaces

P. Holicky, O. Kalenda, L. Vesely +1

On each nonreflexive Banach space X there exists a positive continuous convex function f such that 1/f is not a d.c. function (i.e., a difference of two continuous convex functions…

math.FA2007

On Lipschitz and d.c. surfaces of finite codimension in a Banach space

Luděk Zajíček

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space, and properties of generated $σ$ -ideals are studied. These $σ$ -ideals naturally appear in the diff…

math.CA2006★ 2 cited

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

Jakub Duda, Ludek Zajicek

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 $C^{2}$ parametri…

The Fréchet derivative is in the first Baire class

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

On C1 Whitney extension theorem in Banach spaces

Frechet differentiability via partial Frechet differentiability

On sets of discontinuities of functions continuous on all lines

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

A characterization of sets in R2 with DC distance function

On sets in Rd with DC distance function

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

On the structure of WDC sets

On the structure of sets with positive reach

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

G\^ ateaux and Hadamard differentiability via directional differentiability

Hadamard differentiability via G\^ ateaux differentiability

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

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

Properties of distance functions on convex surfaces and applications

On extensions of d.c. functions and convex functions

Quotients of continuous convex functions on nonreflexive Banach spaces

On Lipschitz and d.c. surfaces of finite codimension in a Banach space

Curves in Banach spaces which allow a C2 parametrization

On $C^{1}$ Whitney extension theorem in Banach spaces

A characterization of sets in $R^{2}$ with DC distance function

On sets in $R^{d}$ with DC distance function

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