arXiv subjects
Publications and source records attributed to Anatolij Plichko.
A negative solution of Problem 188 posed by Max Eidelheit in {\it the Scottish Book} concerning superpositions of separately absolutely continuous functions is presented. We discuss here his and some related problems which have also negative solutions. Finally, we give an explanation of such negative answers from the "embeddings of Banach spaces" point of view.
Let $μ$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $μ$-continuous if $μ(\partial U)=0$. In the paper the $μ$-continuity and uniform $μ$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $μ$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.
We find conditions for a smooth nonlinear map $f:U\rightarrow V$ between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some $c$ and each positive $\varepsilon<c$ the image $% f(B_\varepsilon(x))$ of each $\varepsilon$-ball $B_\varepsilon(x)\subset U$ is convex. We give a lower bound on $c$ via the second order Lipschitz constant $\mathrm{Lip}_2(f)$, the Lipschitz-open constant $\mathrm{Lip}_o(f)$ of $f$, and the 2-convexity number $\mathrm{conv}_2(X)$ of the Banach space $% X$.