  We introduce two notions called $k-$weakly uniform rotundity ($k-$WUR) and
$k-$weakly locally uniform rotundity ($k-$WLUR) in real Banach spaces. These
are natural generalizations of the well-known concepts $k-$UR and WUR. By
introducing two best approximation notions namely $k-$weakly strong
Chebyshevity and $k-$weakly uniform strong Chebyshevity, we generalize some of
the existing results to $k-$WUR and $k-$WLUR spaces. In particular, we present
characterizations of $k-$WUR spaces in terms of $k-$weakly uniformly strong
Chebyshevness. Also, the inheritance of the notions $k-$WUR and $k-$WLUR by
quotient spaces are discussed. Further, we provide a necessary and sufficient
condition for an infinite $\ell_p-$product space to be $k-$WUR (respectively,
$k-$WLUR). As a consequence, we observe that the notions WUR and $k-$WUR
coincide for an infinite $\ell_p-$product of a Banach space.
