Smooth approximation of convex bodies (Q760666)

Extending 2-dimensional results of \textit{S. Tanno} [J. Math. Soc. Japan 28, 384-395 (1976; Zbl 0318.52002)] and \textit{B. Wegner} [ibid. 29, 537- 540 (1977; Zbl 0363.53001)] (see also the reviewer's survey [Convexity and its applications, Collect. Surv., 131-162 (1983; Zbl 0519.52005)] the author proves that each convex body K in \({\mathbb{R}}^ d\) can be approximated arbitrarily closely by convex bodies having an algebraic support function and with everywhere positive radii of curvature. The approximating bodies can be chosen such that they have at least the same group of symmetries as K. For K of constant width one may choose approximating bodies of constant width. Tools for the proof are a kind of convolution and expansion into spherical harmonics.