Anisotropic Caffarelli-Kohn-Nirenberg inequalities for the Minkowski functional: $L^p$-remainder identities and extremals

We establish exact anisotropic $L^p$-Hardy and Caffarelli-Kohn-Nirenberg identities associated with the Minkowski functional $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_K$ and the anisotropic radial derivative ${\mathcal{R}}_K$, where $1<p<\mathrm{\infty }$ and $K\subset \text{RN}$ is a smooth and strictly convex body containing the origin in its interior, not necessarily origin-symmetric. These identities contain explicit nonnegative $R_p$-remainders and yield sharp anisotropic radial $L^p$-Hardy and $L^p$-Caffarelli-Kohn-Nirenberg inequalities. In the noncritical parameter regions, we characterize the full family of extremal functions in the natural weighted completion space, while in the critical case the sharp constant is not attained by any nonzero function in that space. As applications, we establish sharp anisotropic $L^p$-Heisenberg uncertainty principles and obtain explicit anisotropic Gaussian-type optimizers. Finally, letting $K^{\ast }$ denote the polar body of $K$, we derive norm-based anisotropic gradient inequalities associated with the origin-symmetric body $K^{\ast }\cap (-K^{\ast })$. When $K$ is origin-symmetric, the corresponding gradient constants are sharp, including those in the critical case.