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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 \(\int_{\ell_2} (A_{\varepsilon})=0\).

MSC:
46B20 Geometry and structure of normed linear spaces
46G05 Derivatives of functions in infinite-dimensional spaces
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