zbMATH — the first resource for mathematics

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space. (English) Zbl 1034.46016
The authors study the sizes of the sets \({\mathcal C}(b)\) of gradients of bump (i.e., smooth, with bounded support, \(\not\equiv 0\)) functions \(b\) on the Hilbert space \(\ell_2\), and the related question as to how small the set \({\mathcal C}(A)\) of tangent hyperplanes to a smooth bounded starlike body \(A\) in \(\ell_2\) can be. Here \({\mathcal C}(b)=\{\lambda b'(x)\mid x\in X\), \(\lambda \geqslant 0\}\), \({\mathcal C}(A)=\{x^*\in X^* \mid x\in \text{Ker\,} x^*\) is tangential to \(A\) at \(x\in \partial A\}\). This study of topological sizes of \({\mathcal C}(b)\) and \({\mathcal C}(A)\) was initiated in [D. Azagra and R. Deville, J. Funct. Anal. 180, 328–346 (2001; Zbl 0983.46016)].
The first main result of the present paper states, in particular, that there is a \(C^1\) Lipschitz bump function \(b\) on \(\ell_2\) (with Lipschitz derivative) such that the cone \({\mathcal C}(b)\) has empty interior and \(b'(x)\neq 0\) for every \(x\) in the interior of its support. The second main result states that there are \(C^1\) smooth Lipschitz bounded starlike bodies \(A_{\varepsilon}\) in \(\ell_2\), \(0< \varepsilon<1\), such that their Minkowski functionals uniformly approximate the usual \(\ell_2\)-norm on bounded sets and \(\int_{\ell_2} (A_{\varepsilon})=0\).

46B20 Geometry and structure of normed linear spaces
46G05 Derivatives of functions in infinite-dimensional spaces
Full Text: DOI Link