For an arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelogram containing the domain is constructed.

As an application we obtain a new proof that every planar convex domain possesses optimal polynomial meshes.

中文翻译：

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Christoffel 函数在平面凸域上的几何计算

对于任意平面凸域，我们使用与其他简单参考域的比较来计算 Christoffel 函数的行为，直到一个常数因子。下界是通过构造包含在域中的适当椭圆来获得的，而对于上界，构建了包含域的适当平行四边形。

作为一个应用，我们获得了一个新的证明，即每个平面凸域都具有最优多项式网格。