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On a discrete Dido-type question. (English) Zbl 0703.05036

The following two results, concerning a connected simple graph \(G_ n^ d\) with n edges and linearly embedded in the Euclidean d-space, are proved:
Theorem 1: If \(H_ n(n\geq 4)\) is a polygonal n-path inscribed in a hemicircle and its edges are congruent to those of \(G^ 2_ n\), then the area of the convex hull of \(G^ 2_ n\) is smaller than or equal to the area of the convex hull of \(H_ n\); the equality holds if and only if \(G_ n\) is itself a polygonal path inscribed in a hemicircle.
Theorem 2: If \(S_{d+1}(d\geq 2)\) is a \((d+1)\)-star centered at the orthocenter of the simplex spanned by it and its edges are congruent to those of \(G^ d_{d+1}\), then the d-volume of the convex hull of \(G^ d_{d+1}\) is smaller than or equal to the d-volume of the convex hull of \(S_{d+1}\).
Reviewer: J.Weinstein

MSC:

05C38 Paths and cycles
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
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