Abstract
Each convex planar set K has a perimeter C, a minimum width E, an area A, and a diameter D. The set of points (E,C, A1/2, D) corresponding to all such sets is shown to occupy a cone in the non-negative orthant of R4with its vertex at the origin. Its three-dimensional cross section S in the plane D = 1 is investigated. S lies in a rectangular parallelepiped in R3. Results of Lebesgue, Kubota, Fukasawa, Sholander, and Hemmi are used to determine some of the boundary surfaces of S, and new results are given for the other boundary surfaces. From knowledge of S, all inequalities among E, C ,A, and D can be found.
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Ting, L., Keller, J. Extremal Convex Planar Sets. Discrete Comput Geom 33, 369–393 (2005). https://doi.org/10.1007/s00454-004-1145-z
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DOI: https://doi.org/10.1007/s00454-004-1145-z