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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
##### Keywords:
Banach space; smooth bump function; smooth body; starlike bodies
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