• Delta-konvexní zobrazení, Master Thesis, Charles University, Prague, 2000.
• Delta-konvexní zobrazení, Rerum Naturalium Doctoris (RNDr.) Thesis, Charles University, Prague, 2002.
• Aspects of delta-convexity, PhD dissertation, University of Missouri, Columbia, 2003.
• Delta-convexity, metric projection and negligible sets, Ph.D. Thesis, Charles University, Prague, 2004. (Abstract published in Comment. Math. Univ. Carolin. 46 (2005), no.1, 182--183.)

On inverses of $\delta$-convex mappings, Comment. Math. Univ. Carolin. 42 (2001), no. 2, 281--297. (PDF or PS can be obtained here.) MR1832147(2002c:46092)

In the first part of this paper, we prove that in a sense the class of bilipschitz $\delta$-convex mappings, whose inverses are locally $\delta$-convex, is stable under finite-dimensional $\delta$-convex perturbations. In the second part, we construct two $\delta$-convex mappings from $\ell_1$ onto $\ell_1$, which are both bilipschitz and their inverses are nowhere locally $\delta$-convex. The second mapping, whose construction is more complicated, has an invertible strict derivative at $0$. These mappings show that for (locally) $\delta$-convex mappings an infinite-dimensional analogue of the finite-dimensional theorem about $\delta$-convexity of inverse mappings (proved in [VZ] L. Veselý, L. Zajíček, Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) {\textbf{289}} (1989)) cannot hold in general (the case of $\ell_2$ is still open) and answer three questions posed in [VZ].

(with L. Veselý, L. Zajíček) On d.c. functions and mappings, Atti Sem. Mat. Fis. Univ. Modena 51 (2003), no. 1, 111--138. MR1993883(2004f:49030)

A function on a Banach space is called d.c. if it is a difference of two continuous convex functions. A theory of d.c. mappings between Banach spaces (which generalize P. Hartman's notion of d.c. mappings between Euclidean spaces) was built in [VZ] L. Veselý, L. Zajíček, Delta-convex mappings between Banach spaces and applications, Dissertationes Math. (Rozprawy Mat.) 289 (1989). In the introduction we present basic information on d.c. functions and mappings and briefly comment on several recent articles on this topic. In Section 2 we consider the question whether each $C^{1,1}$ function (or mapping) defined on an open convex subset $A$ of a Banach space is d.c. As a consequence of our results we obtain a negative answer to a problem from [VZ]. In the Section 3 we consider the question whether a d.c. mapping defined on an open convex subset $C$ of a Banach space $X$ is Lipschitz if it has a Lipschitz control function. We show that the answer is negative in general but it is positive if $C$ is bounded or if $C$ contains (a translate of) a nonempty open convex cone. In Sections 4 and 5 we present several observations which concern natural questions about d.c. mappings between Banach spaces.

On the size of the set of points where the metric projection exists, Israel J. Math. 140 (2004), 271--283. MR2054848(2005a:46028)

In this paper we answer in the negative a question due to J. P. R. Christensen about almost everywhere existence of nearest points using a decomposition of $\ell_2$ due to J. Matoušek and E. Matoušková. We also formulate a similar question about almost everywhere existence of farthest points and answer it in the negative.

(with L. Zajíček) The Banach-Zarecki Theorem for functions with values in metric spaces, Proc. Amer. Math. Soc. 133 (2005), 3631--3633. MR2163600

Using an old result of Luzin about his property $(N)$, we prove a general version of Banach-Zarecki theorem (on absolute continuity and Luzin's property $(N)$).

(with L. Zajíček) Curves in Banach spaces - differentiability via homeomorphisms, to appear in the Rocky Mountain Journal of Mathematics. (Preprint available from here.)

We prove several results on curves $ f: [0,1] \to X$, where $X$ is an arbitrary real Banach space. They generalize theorems which were proved by Zahorski, Tolstov, Choquet and Bari in the case $X=\R^n$. First we give a complete characterization of those $f$ that admit an equivalent parametrization which has a continuous derivative (resp. with continuous derivative which is non-zero everywhere or almost everywhere). Further we establish theorems characterizing curves allowing boundedly or finitely differentiable parametrizations (with almost everywhere non-zero derivative). As a tool, we prove versions of the aforementioned theorems for metric analogues of derivatives. Finally, we discuss the case of curves allowing almost everywhere differentiable parametrizations. We also answer several questions posed by Bruckner.

On positively differentiable curves with values in normed linear spaces, J. Math. Anal. Appl. 320 (2006), 662--674. Journal link.

We prove a theorem that characterizes continuous normed linear space-valued curves allowing differentiable parameterizations with non-zero derivatives as those curves, all the points of which are regular (in Choquet's sense). We also state an equivalent geometric condition not involving any homeomorphisms. This extends a theorem due to Choquet, who proved a similar result for curves with values in Euclidean spaces.

On the size of the set of points where the metric projection is discontinuous, J. Nonlinear Convex Anal. 7 (2006), 67--70.

We show that the set of points where the metric projection onto a closed set in a separable Hilbert space is single-valued but discontinuous can be covered by countably many d.c.-hypersurfaces. As a corollary, we get a similar result for the metric projection onto a Čebyšev set. This complements a result of Konyagin.

Curves with finite turn, to appear in Czechoslovak Math. J. (Preprint available from here.)

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces -- one of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one- sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results (for example Corollary \ref{exportcor} concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces.

Metric and $w^*$-differentiability of pointwise Lipschitz mappings, to appear in Journal of Analysis and its Applications. (Preprint available from here.)

We study the metric and $w^*$-differentiability of pointwise Lipschitz mappings. First, we prove several theorems about metric and $w^*$-differentiability of pointwise Lipschitz mappings between $R^n$ and a Banach space $X$ (which extend results due to Ambrosio, Kirchheim and others), then apply these to functions satisfying the spherical Rado-Reichelderfer condition, and to absolutely continuous functions of several variables with values in a Banach space. We also establish the area formula for pointwise Lipschitz functions, and for $(n,\lambda)$-absolutely continuous functions with values in Banach spaces. In the second part of this paper, we prove two theorems concerning metric and $w^*$-differentiability of pointwise Lipschitz mappings $f:X\to Y$ where $X,Y$ are Banach spaces with $X$ being separable (resp. $X$ separable and $Y=G^*$ with $G$ separable).

On Gateaux differentiability of pointwise Lipschitz mappings, to appear in the Canadian Mathematical Bulletin. (Preprint available from arXiv.org.)

We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f:X\to\R$ cone monotone, $g:X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Frechet differentiable.

On a decomposition of Banach spaces, to appear in Colloquium Mathematicum. (Preprint available from here.)

By using D. Preiss' approach to a construction from a paper by J. Matoušek and E. Matoušková, and some results of E. Matoušková, we prove that we can decompose a separable Banach space space with modulus of convexity of power type $p$ as a union of a ball small set (in a rather strong symmetric sense), and a set, which is Aronszajn null. This improves an earlier unpublished result of E. Matoušková. As a corollary, we get that in each separable Banach space with modulus of convexity of power type $p$, there exists a closed nonempty set $A$, and a Borel non Haar null set $Q$, such that no point from $Q$ has a nearest point in $A$. Another corollary of our approach is that $\ell_1$ and $L_1$ can be decomposed as a union of a ball small set and an Aronszajn null set. The decomposition results are an improvement of earlier results of D. Preiss and J. Tišer in the appropriate spaces.

Absolutely continuous functions with values in metric spaces, to appear in Real Analysis Exchange. (Preprint available from here.)

We present a general theory of absolutely continuous paths with values in metric spaces using the notion of metric derivatives. Among other results, we prove analogues of the Banach-Zarecki and Vallée Poussin theorems.

Cone monotone mappings: continuity and differentiability, to appear in Nonlinear Analysis: Theory, Methods & Applications. (Preprint available from arXiv.org.)

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces. We define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): $K$-monotone dominated and cone-to-cone monotone mappings. $K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and $K$-monotone functions defined by Borwein, Burke and Lewis, to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zajíček, we show some relationships between these classes. Then, we show that every $K$-monotone function $f:X\to\R$, where $X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for $K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning differentiability (also in metric and $w^*$-senses) of these mappings.

Second order differentiability of paths via a generalized $\frac{1}{2}$-variation, submitted. (Preprint available from arXiv.org.)

We find an equivalent condition for a continuous vector-valued path to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a $VBG_{\frac{1}{2}}$ function, which plays an analogous r\^ole for the second order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability. In fact, for a function $f:[a,b]\to X$, being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function with a pointwise Lipschitz derivative. We also consider the case when the first derivative can be taken non-zero almost everywhere.

Generalized $\alpha$-variation and Lebesgue equivalence to differentiable functions, submitted. (Preprint available from arXiv.org.)

We find an equivalent condition for a real function $f:[a,b]\to\R$ to be Lebesgue equivalent to an $n$-times differentiable function ($n\geq 2$); a simple solution in the case $n=2$ appeared in an earlier paper. For that purpose, we introduce the notions of $CBVG_{1/n}$ and $SBVG_{1/n}$ functions, which play analogous roles for the $n$-th order differentiability as the classical notion of a $VBG_*$ function for the first order differentiability, and the classes $CBV_{1/n}$ and $SBV_{{1}/{n}}$ (introduced by Preiss and Laczkovich) for $C^n$ smoothness. As a consequence of our approach, we obtain that Lebesgue equivalence to $n$-times differentiable function is the same as Lebesgue equivalence to a function $f$ which is $(n-1)$-times differentiable with $f^{(n-1)}(\cdot)$ being pointwise Lipschitz. We also characterize the situation when a given function is Lebesgue equivalent to an $n$-times differentiable function $g$ such that $g'$ is nonzero a.e. As a corollary, we establish a generalization of Zahorski's Lemma for higher order differentiability.

(with L. Zajíček) Curves in Banach spaces which allow a $C^2$ parametrization or a parametrization with finite convexity, submitted. (Preprint available from arXiv.org.)

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space) which allow an equivalent (smooth) parametrization with finite convexity, and (in the case when $X$ has a Fr\'echet smooth norm) a $C^2$ parametrization. For $X=\R$, a characterization for the $C^2$ case is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed.

(with O. Maleva) Metric derived numbers and continuous metric differentiability via homeomorphisms, submitted. (Preprint available from arXiv.org.)

We define the notions of unilateral metric derivatives and ``metric derived numbers'' in analogy with Dini derivatives (also referred to as ``derived numbers'') and establish their basic properties. We also prove that the set of points where a path with values in a metric space with continuous metric derivative is not ``metrically differentiable'' (in a certain strong sense) is $\sigma$-symmetrically porous and provide an example of a path for which this set is uncountable. In the second part of this paper, we study the continuous metric differentiability via a homeomorphic change of variable.

(with L. Zajíček) On vector-valued curves that allow $C^{1,\alpha}$ parametrization, in preparation.

Differentiability of vector-valued curves via homeomorphisms, in preparation.

(with Boaz Tsaban) Games in Banach spaces: Questions and several answers, in preparation. (Preprint available from arXiv.org.)

Implicite functions and properties of the distance function, in preparation.

(with L. Zajíček) On differentiation of semiconcave functions, in preparation.

(with L. Zajíček) On propagation of singularities of semiconcave functions, in preparation.

(with L. Zajíček) Semiconvex functions: extensions and representations as a supremum of a family of smooth functions , in preparation.

(with L. Zajíček) Differentiation of semiconcave functions on small convex sets , in preparation.