Geodesic
Small set in a large box
Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 348-360

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Let $K\subset\mathbb R^d$ be a compact convex set which is an intersection of half-spaces defined by at most two coordinates. Let $Q$ be the smallest axes-parallel box containing $K$. We show that as the dimension $d$ grows, the ratio $\operatorname{diam}Q/\operatorname{diam}K$ can be arbitrarily large. We also give examples of compact sets in Banach spaces, which are not contained in any compact contractive set.
Keywords: convex compact subset of $\mathbb R^d$, axes-parallel box, contractive set, graph, random hypergraph. 
@article{MZM_2007_81_3_a4,
     author = {E. Kopeck\'a},
     title = {Small set in a large box},
     journal = {Matemati\v{c}eskie zametki},
     pages = {348--360},
     publisher = {mathdoc},
     volume = {81},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a4/}
}
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E. Kopecká. Small set in a large box. Matematičeskie zametki, Tome 81 (2007) no. 3, pp. 348-360. http://geodesic.mathdoc.fr/item/MZM_2007_81_3_a4/