Geodesic
On the Lassak conjecture for a convex body
Modelirovanie i analiz informacionnyh sistem, Tome 18 (2011) no. 3, pp. 5-11

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In 1993 M. Lassak formulated (in the equivalent form) the following conjecture. If we can inscribe a translate of the cube $[0,1]^n$ into a convex body $C\subset\mathbb R^n$, then $\sum_{i=1}^n 1/w_i\geq 1$. Here $w_i$ denotes the width of $C$ in the direction of the $i$th coordinate axis. The paper contains a new proof of this statement for $n=2$. Also we show that if a translate of $[0,1]^n$ can be inscribed into the $n$-dimensional simplex, then for this simplex holds $\sum_{i=1}^n 1/w_i= 1$.
Keywords: convex body, width, axial diameter, homothety, projection.
Mots-clés : simplex, interpolation 
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     author = {M. V. Nevskii},
     title = {On the {Lassak} conjecture for a convex body},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {5--11},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2011_18_3_a0/}
}
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M. V. Nevskii. On the Lassak conjecture for a convex body. Modelirovanie i analiz informacionnyh sistem, Tome 18 (2011) no. 3, pp. 5-11. http://geodesic.mathdoc.fr/item/MAIS_2011_18_3_a0/