Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces.

*(English)*Zbl 1032.46097We use the authors’ text from the introduction of the paper with some modifications.

It is well-known that in a finite-dimensional Banach space there is no continuous retraction from the unit ball onto the unit sphere. This fact is equivalent to Brouwer’s fixed point theorem. This result is no longer true in infinite dimensions. B. Nowak [Bull. Acad. Pol. Sci., Ser. Sci. Math. 27, 861-864 (1979; Zbl 0472.54008)] showed that for several infinite-dimensional Banach spaces, Brouwer’s theorem fails, even for Lipschitz mappings. In [Proc. Am. Math. Soc. 88, 439-445 (1983; Zbl 0518.46010)], Y. Benyamini and Y. Sternfeld generalized Nowak’s result for all infinite-dimensional Banach spaces, establishing that for every infinite-dimensional space \({(X,\|\cdot\|)}\) there exists a Lipschitz retraction from the unit ball onto the unit sphere.

In recent years, a lot of work has been done on smoothness and Lipschitz properties in Banach spaces. Following this trend, the authors show that the Nowak-Benyamini-Sternfeld results can be sharpened so as to get \(C^p\) smooth Lipschitz retractions of the unit ball onto the unit sphere in every infinite-dimensional Banach space with \(C^p\) smooth norm. In fact, these results are obtained for a wider class of objects than balls and spheres. It is shown that for every infinite-dimensional Banach space with a \(C^p\) Lipschitz bounded starlike body \(A\) \((p=0,1,2,\dots)\), there is \(C^p\) Lipschitz retraction of \(A\) onto its boundary \(\partial A\), and \(\partial A\) is \(C^p\) Lipschitz contractible.

It is well-known that in a finite-dimensional Banach space there is no continuous retraction from the unit ball onto the unit sphere. This fact is equivalent to Brouwer’s fixed point theorem. This result is no longer true in infinite dimensions. B. Nowak [Bull. Acad. Pol. Sci., Ser. Sci. Math. 27, 861-864 (1979; Zbl 0472.54008)] showed that for several infinite-dimensional Banach spaces, Brouwer’s theorem fails, even for Lipschitz mappings. In [Proc. Am. Math. Soc. 88, 439-445 (1983; Zbl 0518.46010)], Y. Benyamini and Y. Sternfeld generalized Nowak’s result for all infinite-dimensional Banach spaces, establishing that for every infinite-dimensional space \({(X,\|\cdot\|)}\) there exists a Lipschitz retraction from the unit ball onto the unit sphere.

In recent years, a lot of work has been done on smoothness and Lipschitz properties in Banach spaces. Following this trend, the authors show that the Nowak-Benyamini-Sternfeld results can be sharpened so as to get \(C^p\) smooth Lipschitz retractions of the unit ball onto the unit sphere in every infinite-dimensional Banach space with \(C^p\) smooth norm. In fact, these results are obtained for a wider class of objects than balls and spheres. It is shown that for every infinite-dimensional Banach space with a \(C^p\) Lipschitz bounded starlike body \(A\) \((p=0,1,2,\dots)\), there is \(C^p\) Lipschitz retraction of \(A\) onto its boundary \(\partial A\), and \(\partial A\) is \(C^p\) Lipschitz contractible.

Reviewer: Victor Milman (Minsk)