Growth properties of entire functions depending on a parameter
We study the growth of parameter-dependent entire functions. We are mainly interested in the case where the functions depend holomorphically on the parameter.
How many intervals cover a point in random dyadic covering?
Least deviation decomposition with respect to a pair of convex sets.
Lipschitz sums of convex functions
We give a geometric characterization of the convex subsets of a Banach space with the property that for any two convex continuous functions on this set, if their sum is Lipschitz, then the functions must be Lipschitz. We apply this result to the theory of Δ-convex functions.
Logarithmic functional mean in convex analysis.
Maximizing the Bregman divergence from a Bregman family
The problem to maximize the information divergence from an exponential family is generalized to the setting of Bregman divergences and suitably defined Bregman families.
Monotone Valuations on the Space of Convex Functions
We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.
More than first-order developments of convex functions: primal-dual relations.
Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum
We construct a Lipschitz function on which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.
On local convexity of nonlinear mappings between Banach spaces
We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ɛ < c the image f(B ɛ(x)) of each ɛ-ball B ɛ(x) ⊂ U is convex. We give a lower bound on c via the second order Lipschitz constant Lip2(f), the Lipschitz-open constant Lipo(f) of f, and the 2-convexity number conv2(X) of the Banach space X.
On monotone minimal cuscos
On the characterization of some families of closed convex sets.
On the continuity of functions.
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in -dimensional Euclidean space . It is proved that if , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, covariant, and associative if and only if it is addition for some . It is also demonstrated that if ,...
Outer -convex functions on a normed space.
Parallel synchronous algorithm for nonlinear fixed point problems.
p-convex functions in linear spaces
Point derivations for Lipschitz functions andClarke's generalized derivative
Clarke’s generalized derivative is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed and fixed the function is continuous and sublinear in . It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.
Schur-convexity for a class of symmetric functions and its applications.
Separation by hyperplanes in finite-dimensional vector spaces over Archimedean ordered fields.