Abstract
We prove the following isoperimetric inequality in R2, conjectured by Glazyrin and Morić. Given a convex body S and k pairwise disjoint convex bodies C 1,...,C k that are contained in S, then Σ k i=1 Per(C i)≤Per(S)+2(k—1)Diam(S). Here Per(.) denotes the perimeter of a set and Diam(.) is the diameter of a set.
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References
A. Glazyrin and F. Morić: Upper bounds for the perimeter of plane convex bodies, Acta Math. Hungar. 142 (2014), 366–383.
G. B. Price: On the completeness of a certain metric space with an application to Blaschke’s selection theorem, Bull. Amer. Math. Soc. 46 (1940), 278–280.
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This research was supported by ISF grant (grant No. 1357/12)
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Pinchasi, R. On the perimeter of k pairwise disjoint convex bodies contained in a convex set in the plane. Combinatorica 37, 99–125 (2017). https://doi.org/10.1007/s00493-015-3217-5
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DOI: https://doi.org/10.1007/s00493-015-3217-5