About estimates of the $\mathscr{K}$-divisibility constant for Banach pairs

The paper contains some results on estimates of the $\mathscr{K}$-divisibility constant for Banach pairs. Its has been established that it is impossible to improve the estimate $3+2\sqrt2$ for any Banach pair and $4$ any pair of Banach lattices using the method of Yu. A. Brudnyi and N. Ya. Krugljak. I give a proof of Sedaev–Semenov theorem for the pair $(L_1^1,L_1)$ with measure on half-axis, using only the properties of concave functions.