Nodal Sets of Smooth Functions with Finite Vanishing Order and p-Sweepouts

We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linear combinations of any $p+1$ smooth functions form an admissible $p-$sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques--Neves upper bounds on the min-max $p$-widths of $M.$ We also prove that close to a point at which a smooth function on $\mathbb{R}^{n+1}$ vanishes to order $k$, its nodal set is contained in the union of $k$ $W^{1,p}$ graphs for some $p > 1$. This implies that the nodal set is locally countably $n$-rectifiable and has locally finite $\mathcal{H}^n$ measure, facts which also follow from a previous result of B\"{a}r. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.