Banach 空间中的 Birkhoff 正交 Dehghan--Rooin 型常数

中文摘要 AI 辅助

设 $X$ 为实 Banach 空间。Dehghan--Rooin 量比较角距离 $\alpha[x,y]=\\|x/\\|x\\|-y/\\|y\\|\\|$ 与斜角距离 $\beta[x,y]=\\|x/\\|y\\|-y/\\|x\\|\\|$。受 Birkhoff 正交性在 Massera--Schaffer 不等式中的近期应用的启发，我们引入并研究 Birkhoff 限制的 Dehghan--Rooin 常数 \\[ \operatorname{DR}_{B}(X)=\sup\left\{\frac{\alpha[x,y]}{\beta[x,y]}:x,y\in X\setminus\{0\},\\ x\perp_{B} y,\\ \beta[x,y]\ne0\right\}. \\] 限制条件 $x\perp_{B} y$ 将 $\alpha$ 与 $\beta$ 的全局比较转化为一个方向性不变量，该不变量检测支撑泛函与径向归一化映射之间的相互作用。我们获得了尺度轮廓公式、锐利初等界、子空间和超幂下的稳定性、极值情形 $\operatorname{DR}_{B}(X)=1$ 的变分刻画、用经典 Dehghan--Rooin 常数给出的估计，以及对一致凸和一致光滑空间的推论。在光滑二维空间中，我们通过归一化对偶映射推导出显式公式；在 Radon 平面中，我们将常数简化为单位圆上一段弧上的一维优化。文中包含若干模型计算，包括 Hilbert 空间、多面体平面和二维 $p$-范数空间。最后一节通过伴随的 Birkhoff 角模量给出正规结构类型的推论，并列出由新常数引发的若干问题。

英文摘要

For a real Banach space $X$. The Dehghan--Rooin magnitude compares the angular distance $α[x,y]=\|x/\|x\|-y/\|y\|\|$ together with the skew angular distance $β[x,y]=\|x/\|y\|-y/\|x\|\|$. Motivated by the recent use of Birkhoff orthogonality in the Massera--Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan--Rooin constant \[ \operatorname{DR}_{B}(X)=\sup\left\{\frac{α[x,y]}{β[x,y]}:x,y\in X\setminus\{0\},\ x\perp_{B} y,\ β[x,y]\ne0\right\}. \] The restriction $x\perp_{B} y$ turns the global comparison of $α$ and $β$ into a directional invariant which detects the interaction between supporting functionals and the radial normalization map. We obtain a scale-profile formula, sharp elementary bounds, stability under subspaces and ultrapowers, a variational characterization of the extremal case $\operatorname{DR}_{B}(X)=1$, estimates in terms of the classical Dehghan--Rooin constant, and consequences for uniformly convex and uniformly smooth spaces. In smooth two-dimensional spaces we derive an explicit formula through the normalized duality mapping, and in Radon planes we reduce the constant to a one-dimensional optimization on an arc of the unit circle. Several model computations are included, including Hilbert spaces, polyhedral planes, and two-dimensional $p$-norm spaces. The final section gives normal-structure type consequences through a companion Birkhoff angular modulus and lists several problems suggested by the new constant.

发表机构

• School of Mathematics and Statistics, Anqing Normal University（安庆师范大学数学与统计学院）

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