Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12.

中文翻译：

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值域为圆盘的矩阵处数值半径的部分平滑

涉及数值半径的优化问题的解决方案通常属于一个特殊类别：具有值域的矩阵集，圆盘以原点为中心。在用一些例子说明了这种现象之后，我们通过研究矩阵来阐明它，这组“磁盘矩阵”是围绕其数值半径部分平滑的流形。然后，我们将我们的结果应用于非零点由单个超对角线组成的矩阵，例如 Jordan 块和与著名的 Crouzeix 猜想相关的 Crabb 矩阵。最后，我们考虑任意复杂的三乘三矩阵；即使在这种情况下，细节也出奇地复杂。我们的一个结果是，在这个维度为 18 的实向量空间中，盘矩阵集是维度为 12 的半代数流形。