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Reduced strictly convex plane figure is of constant width. (English) Zbl 0586.52002
A convex body \(K\subset R^ d\) is called reduced if for each convex body K’\(\subset K (K'\neq K)\) the width of K’ is less than the width of K. In 1978 W. Heil posed the problem if each strictly convex reduced body is of constant width. For reduced convex bodies with smooth boundary this was proved by H. Groemer [Monatsh. Math. 96, 29-39 (1983; Zbl 0513.52003)]. The author proves Heil’s problem for \(d=2\). A solution of Heil’s problem for all d is announced by J. Böhm [Diskrete Geometrie, 3. Kolloq., Salzburg 1985, 49-51 (1985; Zbl 0587.52001)].
Reviewer: J.M.Wills

MSC:
52A10 Convex sets in \(2\) dimensions (including convex curves)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A40 Inequalities and extremum problems involving convexity in convex geometry
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[1] BONNESEN, T. and FENCHEL, W.; Theorie der konvexen Körper; Chelsea Publishing Co., Bronx, New York. · Zbl 0008.07708
[2] GROEMER, H., Extremal convex Sets; Monatshefte für Mathematik 96 (1983), 29-39. · Zbl 0513.52003
[3] GRUBER, P. M. and SCHNEIDER, R.; Problems in Geometric Convexity. In: Contributions to Geometry. Proc. Geom. Sympos., Siegen 1978. Basel-Boston-Stuttgart, Birkaüser, 1979, 258-278.
[4] HEIL, E., Kleinste konvexe Körper gegebener Dicke. Preprint #453, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1978.
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