Let $\Omega$ be an open convex set in $\mathbb{R}^m$ with finite width, and with boundary $\partial \Omega$. Let $v_{\Omega}$ be the torsion function for $\Omega$, i.e., the solution of $-\Delta v=1, v|_{\partial\Omega}=0$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge {\pi^2}/{24}$, and that this bound is sharp.

中文翻译：

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在用于菱形，等腰三角形和稀疏凸集的Pólya函数上

假设$ \ Omega $是$ \ mathbb {R} ^ m $中的一个开放凸集，其宽度有限，边界为$ \ partial \ Omega $。令$ v _ {\ Omega} $为$ \ Omega $的扭转函数，即$-\ Delta v = 1，v | _ {\ partial \ Omega} = 0 $的解。为$ \ Vert v _ {\ Omega} \ Vert_ {L ^ {\ infty}（\ Omega）} \ lambda（\ Omega）$的乘积获得一个上限，其中$ \ lambda（\ Omega）$为Dirichlet拉普拉斯算子在$ L ^ 2（\ Omega）$中的频谱底部。在凸集的稀疏序列的范围内，上限是尖锐的。对于面积为$ 1 $的平面菱形和等腰三角形，表明$ \ Vert v _ {\ Omega} \ Vert_ {L ^ {1}（\ Omega）} \ lambda（\ Omega）\ ge {\ pi ^ 2} / {24} $，并且这个界限很明显。