  Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at
an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to
investigate the behavior of the polynomial interpolants at the Fej\'er points,
defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of
$\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er
points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev
points and that the Chebyshev points are most commonly used as nodes for
Lagrange polynomial interpolation. On the other hand, numerical experimentation
demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue
constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial
degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er
points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc.
Further numerical experiments also demonstrate that the least upper bounds of
the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation
polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some
$\beta >0$ that depends on $p >1$.
