Orders of moduli of continuity of operators of almost best approximation

Let $X$ be a normed linear space, $Y\subset X$ a finite-dimensional subspace, $\varepsilon>0$. A multiplicative $\varepsilon$-selection $M\colon K\to Y$, where $K\subset X$, is a single-valued mapping such that
$$ \forall\,x\in K\qquad \|Mx-x\|\leqslant\inf\{\|x-y\|:y\in Y\}\cdot(1+\varepsilon). $$

It is proved in the paper that when $X=L^p(T,\Sigma,\mu)$, $1<p<\infty$, for any $Y\subset X$ and $\varepsilon>0$ there exists an $\varepsilon$-selection $M\colon K\to Y$ such that
$$ \forall\,x_1,x_2\in K\qquad \|Mx_1-Mx_2\|\leqslant c(n,p)(1+\varepsilon^{-|1/2-1/p|})\|x_1-x_2\|, $$
where the estimate is order-sharp in the space $L^p[0,1]$. It is also established that the Lipschitz constant for the $\varepsilon$-selection is of proximate order $1/\varepsilon$ in the spaces $L^1[0,1]$ and $C[0,1]$.