  The first part of this paper complements previous results on characterization
of polynomials of least deviation from zero in Sobolev $p$-norm ($1<p<\infty$)
for the case $p=1$. Some relevant examples are indicated.
  The second part deals with the location of zeros of polynomials of least
deviation in discrete Sobolev $p$-norm. The asymptotic distribution of zeros is
established on general conditions. Under some order restriction in the discrete
part, we prove that, the $n$-th polynomial of least deviation has at least
$n-\mathbf{d}^*$ zeros on the convex hull of the support of the measure, where
$\mathbf{d}^*$ denotes the number of terms in the discrete part.
