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A remark on the convex sets of the torus. (English) Zbl 0535.52006

Let T denote the d-dimensional torus \(E^ d/{\mathbb{Z}}\), endowed with the quotient-metric \(\delta(x,y)=\inf \{| x-z|:z\equiv ymod {\mathbb{Z}}^ d\}\). A subset C of T is called convex if any two of its points can be connected with a geodesic segment contained in C. Let \({\mathfrak C}\) denote the space of all closed convex sets in T, endowed with the symmetric difference metric \(\vartheta\). It is shown that the isometries of (\({\mathfrak C},\vartheta)\) into itself are precisely the mappings of the form \(C\to i(C)\) for \(C\in {\mathfrak C}\) where i is an isometry of \((T,\delta)\). This interesting theorem complements results of the reviewer [Mathematika 25, 270-278 (1978; Zbl 0403.52002)], the reviewer and G. Lettl [Bull. Lond. Math. Soc. 12, 455-460 (1980; Zbl 0447.52004)], G. Lettl [(Arch. Math. 35, 471-475 (1980; Zbl 0441.52002)], the reviewer and the author [(Monatsh. Math. 93, 116-126 (1982; Zbl 0449.52001)] and the reviewer [Isr. J. Math. 42, 277-283 (1982; Zbl 0502.52006)].
Reviewer: P.Gruber

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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