Let $$\lambda _1({\tilde{R}}_{2n+1})$$ be the first Dirichlet eigenvalues of the Laplacian associated to regular Reuleaux polygons $${\tilde{R}}_{2n+1}$$ of width $${\tilde{w}}_{2n+1}=1$$ , $$n=1,2,3,\ldots $$ Let $$T({\tilde{R}}_{2n+1})$$ be their torsional rigidities. It is established that $$\lambda _1({\tilde{R}}_{2n+1})$$ is a decreasing sequence, and $$T({\tilde{R}}_{2n+1})$$ an increasing sequence. Similar (but reversed) inequalities hold in the class of regular Reuleaux polygons circumscribed to the unit circle.

中文翻译：

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正则鲁洛多边形的特征值和扭转刚度

令 $$\lambda _1({\tilde{R}}_{2n+1})$$ 是与规则 Reuleaux 多边形相关的拉普拉斯算子的第一个狄利克雷特征值 $${\tilde{R}}_{2n+1 }$$ 宽度 $${\tilde{w}}_{2n+1}=1$$ , $$n=1,2,3,\ldots $$ Let $$T({\tilde{R} }_{2n+1})$$ 是它们的扭转刚度。成立$$\lambda _1({\tilde{R}}_{2n+1})$$是递减序列，$$T({\tilde{R}}_{2n+1}) $$ 一个递增的序列。类似（但相反）的不等式在外接到单位圆的规则 Reuleaux 多边形类中成立。