  Many nonlocal models have adopted Euclidean balls as the nonlocal interaction
neighborhoods. When solving them numerically, it is sometimes convenient to
adopt polygonal approximations of such balls. A crucial question is, to what
extent such approximations affect the nonlocal operators and the corresponding
solutions. While recent works have analyzed this issue for a fixed horizon
parameter, the question remains open in the case of a small or vanishing
horizon parameter, which happens often in many practical applications and has
significant impact on the reliability and robustness of nonlocal modeling and
simulations. In this work, we are interested in addressing this issue and
establishing the convergence of the nonlocal solutions associated with
polygonally approximated interaction neighborhoods to the local limit of the
original nonlocal solutions. Our finding reveals that the new nonlocal solution
does not converge to the correct local limit when the number of sides of
polygons is uniformly bounded. On the other hand, if the number of sides tends
to infinity, the desired convergence can be established. These results may be
used to guide future computational studies of nonlocal models.
