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On the convex bodies of constant width in \(E_n\). (Spanish) Zbl 0063.06712

This is a frequently cited integral-geometric study of convex hypersurfaces of constant width in euclidean \(n\)-space, providing a good example of the author’s excellent and motivating style of writing. It had been published shortly after the end of WW2, when Zentralblatt was in bad shape. Hence in spite of the good quality of the paper so far it had been mentioned in Zentralblatt by title only.
The work is based on the famous survey on convexity given by the book of T. Bonnesen and W. Fenchel [Theorie der konvexen Körper (1934; Zbl 0008.07708), translated into English: Theory of convex bodies. BCS Associates, Moscow, ID (1987; Zbl 0628.52001)]. The basic tool is a formula given in Bonnesen-Fenchel which provides an expression of the volume of an outer parallel hypersurface at distance \(h\) to a convex hypersurface in euclidean \(n\)-space in terms integrals of the elementary symmetric functions of the principal radii of curvature of the given hypersurface and the distance \(h\). Several interpretations of this formula are presented and extensions to inner parallel hypersurfaces are investigated. In the case of a convex hypersurface bounding a convex body of constant width \(a\) the inner parallel hypersurface at distance \(a\) coincides with the given hypersurface. This leads to a series of relations between the integral invariants mentioned above. Some of them can be concluded from others. But, as the author shows, more or less half of them are independent. In the case of a planar curve of constant width \(a\) we get the well-known relation \(L = \pi \cdot a\), \(L\) denoting the length of the curve.

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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