Geodesic
An estimate for the measure of nonconvexity in the $L^p$-space
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 141-145 Cet article a éte moissonné depuis la source Math-Net.Ru

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The measure $\alpha(A)$ of nonconvexity for a bounded subset $A$ of a normed linear space $L$ is the Hausdorff distance between $A$ and its convex hull co $A$. It is proved that if $L$ is an $L^p$-space, then $\alpha(A)\le d(A)/2^{t_p}$, where $d(A)$ is the diameter of $A$ and $t_p=\min\{1/p,1-1/p\}$, $1\le p\le\infty$.Furthermore, this estimate is sharp. 
@article{ZNSL_2001_280_a6,
     author = {N. M. Gulevich and O. N. Gulevich},
     title = {An estimate for the measure of nonconvexity in the $L^p$-space},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {141--145},
     year = {2001},
     volume = {280},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/}
}
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N. M. Gulevich; O. N. Gulevich. An estimate for the measure of nonconvexity in the $L^p$-space. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 7, Tome 280 (2001), pp. 141-145. http://geodesic.mathdoc.fr/item/ZNSL_2001_280_a6/