In this work, we generalize the von Neumann and Davis theorems on the extension of the convexity of a group-invariant norm (resp. function) from the subspace of diagonal matrices to the whole space of complex (resp. hermitian) matrices. Our generalizations go in two directions: (1) We replace the space of complex (hermitian) matrices and its subspace of diagonal matrices by an Eaton triple and its subsystem. (2) We replace the Jensen (J) functional inducing the (usual) convexity of a function by the Hardy–Littlewood–Pólya–Karamata (HLPK) functional related to Wright-convexity and by the Sherman (S) functional connected with Sherman type convexity. The obtained results are interpreted for some classes of matrices.

中文翻译：

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伊顿三元组上 HLPK 和谢尔曼泛函的 Von Neumann-Davis 型定理

在这项工作中，我们将 von Neumann 和 Davis 定理推广到群不变范数（对应函数）的凸性从对角矩阵的子空间扩展到复数（对应的厄密）矩阵的整个空间。我们的推广有两个方向：（1）我们用伊顿三元组及其子系统替换复（厄米）矩阵的空间及其对角矩阵的子空间。(2) 我们用与 Wright 凸性相关的 Hardy-Littlewood-Pólya-Karamata (HLPK) 泛函和与 Sherman 类型相关的 Sherman (S) 泛函替换了诱导函数（通常）凸性的 Jensen (J) 泛函凸性。对某些类别的矩阵解释获得的结果。