The uniqueness of the best non-symmetric $L_1$-approximant for continuous functions with values in $\mathbb{R}^m_p$

The article considers the questions of the uniqueness of the best non-symmetric $L_1$-approximations of continuous functions with values in $\mathbb{R}^m_p, p\in (1;+\infty )$ by elements of the two-dimensional subspace  $H_2= \mathrm{span} \{1, g_{a,b}\}$, where $$g_{a,b}(x)=\left\{ \begin{matrix} -b\cdot (x-1)^2, & x\in [0;1), & \\0, & x\in [1;a-1), & (a\geq 2, b>0),\\(x-a+1)^2,& x\in [a-1,a],&\end{matrix} \right.$$It is obtained that when $b\in (0;1)\cup (1;+\infty), a\geq 2$, the subspace $H_2$ is a unicity space of the best $(\alpha ,\beta )$-approximations for continuous on the $[0;a]$ functions with values in the space  $\mathbb{R}^m_p, p\in (1;+\infty )$. In case $b=1$, $a\geq 4$ it is proved that the subspace  $H_2$ is not a unicity subspace of the best non-symmetric approximations for these functions.Received results summarize the previously obtained Strauss results for the real functions in the case $\alpha = \beta = 1$, as well as the results of Babenko and Glushko for the the best $(\alpha ,\beta )$-approximation for continuous functions on a segment with values in the space $\mathbb{R}^m_p, p\in (1;+\infty )$.