On the Lassak conjecture for a convex body

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 $\sum_{i=1}^n 1/w_i= 1$.