We show that on a compact Riemmanian manifold $(M,g)$, nodal sets of linearcombinations of any $p+1$ smooth functions form an admissible $p-$sweepoutprovided these linear combinations have uniformly bounded vanishing order. Thisapplies in particular to finite linear combinations of Laplace eigenfunctions.As a result, we obtain a new proof of the Gromov, Guth, Marques--Neves upperbounds on the min-max $p$-widths of $M.$ We also prove that close to a point atwhich a smooth function on $\mathbb{R}^{n+1}$ vanishes to order $k$, its nodalset is contained in the union of $k$ $W^{1,p}$ graphs for some $p > 1$. Thisimplies that the nodal set is locally countably $n$-rectifiable and has locallyfinite $\mathcal{H}^n$ measure, facts which also follow from a previous resultof B\"{a}r. Finally, we prove the continuity of the Hausdorff measure of nodalsets under heat flow.
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机译：我们证明，在紧凑的黎曼流形$（M，g）$上，任何$ p + 1 $光滑函数的线性组合的结点集形成一个可允许的$ p- $ sweepout，前提是这些线性组合具有一致有界的消失序。这尤其适用于拉普拉斯特征函数的有限线性组合，因此，我们获得了Gromov，Guth，Marques-Neves的最小-最大$ p $-宽度$ M上界的新证明。接近$ \ mathbb {R} ^ {n + 1} $上的光滑函数对订购$ k $的点消失时，其节点集包含在$ k $ $ W ^ {1，p} $图的并集中对于$ p> 1 $。这意味着该节点集在本地可校正的数量为$ n $，并且具有本地有限的$ \ mathcal {H} ^ n $度量，这一事实也来自先前的B \“ {a} r结果。热流作用下的节点集的Hausdorff测度。
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