  We use Toponogov's triangle comparison theorem from Riemannian geometry along
with quantitative scale oriented variants of classical propagation of
singularities arguments to obtain logarithmic improvements of the
Kakeya-Nikodym norms introduced in \cite{SKN} for manifolds of nonpositive
sectional curvature. Using these and results from our paper \cite{BS15} we are
able to obtain log-improvements of $L^p(M)$ estimates for such manifolds when
$2<p<\tfrac{2(n+1)}{n-1}$. These in turn imply $(\log\lambda)^{\sigma_n}$,
$\sigma_n\approx n$, improved lower bounds for $L^1$-norms of eigenfunctions of
the estimates of the second author and Zelditch~\cite{SZ11}, and using a result
from Hezari and the second author~\cite{HS}, under this curvature assumption,
we are able to improve the lower bounds for the size of nodal sets of Colding
and Minicozzi~\cite{CM} by a factor of $(\log \lambda)^{\mu}$ for any
$\mu<\tfrac{2(n+1)^2}{n-1}$, if $n\ge3$.
