OM KONVEXA KURVOR

Given a closed convex curve of length L and maximum radius of curvature R, the following inequalities are proved: $\mathrm{D}\leqq 2\mathrm{R}\mathrm{sin}\frac{\mathrm{L}}{4\mathrm{R}}$, $\mathrm{d}\geqq 2\mathrm{R}(1-\mathrm{cos}\frac{\mathrm{L}}{4\mathrm{R}})$, $\mathrm{F}\geqq \frac{\mathrm{L}\ma...

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Bibliographic Details
Published in:Nordisk matematisk tidskrift 1955-01, Vol.3 (1), p.57-63
Main Author: PLEIJEL, ARNE
Format: Article
Language:Danish
Online Access:Get full text
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Summary:Given a closed convex curve of length L and maximum radius of curvature R, the following inequalities are proved: $\mathrm{D}\leqq 2\mathrm{R}\mathrm{sin}\frac{\mathrm{L}}{4\mathrm{R}}$, $\mathrm{d}\geqq 2\mathrm{R}(1-\mathrm{cos}\frac{\mathrm{L}}{4\mathrm{R}})$, $\mathrm{F}\geqq \frac{\mathrm{L}\mathrm{R}}{2}-{\mathrm{R}}^{2}\mathrm{sin}\frac{\mathrm{L}}{2\mathrm{R}}$. Here F is the area enclosed by the curve, and D and d are the greatest and the smallest distance between parallel tangents. Equality in each case is attained only for the lens-shaped curve composed of two circular arcs of radius R. If also a lower limit r is given for the radius of curvature, it is shown that the isoperimetric deficit Δ=L2–4πF≦π(4–π)(R–r)2. The upper limit is the best possible, attained by the outer parallel-curve (at a distance r) to a lens with radius R–r and perimeter π(R–r). This is an improvement of an earlier result by Bottema.
ISSN:0029-1412