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We use Toponogov's triangle comparison theorem from Riemannian geometry alongwith quantitative scale oriented variants of classical propagation ofsingularities arguments to obtain logarithmic improvements of theKakeya-Nikodym norms introduced in \cite{SKN} for manifolds of nonpositivesectional curvature. Using these and results from our paper \cite{BS15} we areable 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 ofthe estimates of the second author and Zelditch~\cite{SZ11}, and using a resultfrom 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 Coldingand Minicozzi~\cite{CM} by a factor of $(\log \lambda)^{\mu}$ for any$\mu<\tfrac{2(n+1)^2}{n-1}$, if $n\ge3$.

Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions