Let \(X\) be the space of Lebesgue measurable real functions defined on \([-1,1]\). For \(h\in X\) and \(0<\varepsilon\leq 1\), denote \[ \| h\|_{p,\varepsilon}= \Biggl(\int^\varepsilon_{-\varepsilon}|h(x)|^p dx\Biggr)^{1/p}, \] \(1\leq p<\infty\). Let \(\pi^n\subset X\) be the space of polynomials of degree at most \(n\). Given \(h_i\in X\), \(1\leq i\leq k\), consider the norm \[ P_{p,\varepsilon}(h_1,\dots, h_k)= \sum^k_{i=1}\| h_i\|_{p,\varepsilon}. \] An element \(P_\varepsilon\in \pi^n\) is called a \(P_{p,\varepsilon}\)-best simultaneous approximation (\(P_{p,\varepsilon}\)-b.s.a.) in \(\pi^n\) of the functions \(f_i\in X\), \(1\leq i\leq k\), respect to \(P_{p,\varepsilon}\), if \[ P_{p,\varepsilon}(f_1- P_\varepsilon,\dots, f_k- P_\varepsilon)= \underset{Q\in\pi^n}{}{\text{inf}} P_{p,\varepsilon}(f_1- Q,\dots, f_k-Q). \] In this paper, the authors prove that if \(1< p<\infty\), any \(P_{p,\varepsilon}\)-b.s.a. in \(\pi^n\) of two continuous functions \(f\) and \(g\) in \(X\), interpolates some convex combination of \(f\) and \(g\) in at least \(n+1\) points. If \(p= 2\), a similar result is obtained for \(P_{2,\varepsilon}\)-b.s.a. of \(k\) continuous functions. For \(p= 1\) other necessary conditions over the \(P_{1,\varepsilon}\)-b.s.a. of \(k\) continuous functions is established. An example is given, showing that, in general, the set of cluster points of \(P_\varepsilon\), \(\varepsilon\to 0\) is not unitary even if we have uniqueness of the \(P_{p,\varepsilon}\)-b.s.a. for each \(\varepsilon> 0\). for \(1<p<\infty\), \(k=2\) or \(p= 2\), \(k\geq 2\), it is proved that the set of cluster points of \(P_\varepsilon\), as \(\varepsilon>0\), is a compact and convex set in \(\pi^n\) with the uniform norm.