A C1 function which is nowhere strongly paraconvex and nowhere semiconcave
O třináctém Hilbertově problému
Obecná teorie derivování funkcí a měr na katedře matematické analýzy MFF UK
A Lipschitz function which is on a.e. line need not be generically differentiable
We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...
ISTAM 87 - Bělehrad
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 propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for , these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) , , where , are convex and Lipschitz on . In other words: singularities propagate along arcs with finite turn.
On the differentiation of convex functions in finite and infinite dimensional spaces
Smallness of sets of nondifferentiability of convex functions in non-separable Banach spaces
Alternative definitions of -density topology
Sets of -porosity and sets of -porosity
A note on norm-attaining functionals
Poznámky k teorii integrálu
O Baireových třídách
On differentiability properties of Lipschitz functions on a Banach space with a Lipschitz uniformly Gâteaux differentiable bump function
We improve a theorem of P.G. Georgiev and N.P. Zlateva on Gâteaux differentiability of Lipschitz functions in a Banach space which admits a Lipschitz uniformly Gâteaux differentiable bump function. In particular, our result implies the following theorem: If is a distance function determined by a closed subset of a Banach space with a uniformly Gâteaux differentiable norm, then the set of points of at which is not Gâteaux differentiable is not only a first category set, but it is even -porous...
Singular points of order of Clarke regular and arbitrary functions
Let be a separable Banach space and a locally Lipschitz real function on . For , let be the set of points , at which the Clarke subdifferential is at least -dimensional. It is well-known that if is convex or semiconvex (semiconcave), then can be covered by countably many Lipschitz surfaces of codimension . We show that this result holds even for each Clarke regular function (and so also for each approximately convex function). Motivated by a resent result of A.D. Ioffe, we prove...
Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions
We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on -almost everywhere Fréchet differentiability of Lipschitz functions on (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at -almost every at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are -almost everywhere Fréchet differentiable....
A remark on functions continuous on all lines
We prove that each linearly continuous function on (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 on an arbitrary Banach space , if has moreover the Baire property. We also prove (extending a known finite-dimensional result) that such on a separable is continuous at all points outside a first category set which is also null in any usual sense.
A note on intermediate differentiability of Lipschitz functions
Let be a Lipschitz function on a superreflexive Banach space . We prove that then the set of points of at which has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces ), but it is even -porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.
On the points of multivaluedness of metric projections in separable Banach spaces
Cluster sets of arbitrary functions in Euclidean spaces (Preliminary communication)
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