Convexity Conditions of Kantorovich Function and Related Semi-infinite Linear Matrix Inequalities

The Kantorovich function $(x^TAx)(x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From matrix/convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we investigate the convexity of this function by the condition number of its matrix. In 2-dimensional space, we prove that the Kantorovich function is convex if and only if the condition number of its matrix is bounded above by $3+2\sqrt{2},$ and thus the convexity of the function with two variables can be completely characterized by the condition number. The upper bound `$3+2\sqrt{2}$' is turned out to be a necessary condition for the convexity of Kantorovich functions in any finite-dimensional spaces. We also point out that when the condition number of the matrix (which can be any dimensional) is less than or equal to $\sqrt{5+2\sqrt{6}},$ the Kantorovich function is convex. Furthermore, we prove that this general sufficient convexity condition can be remarkably improved in 3-dimensional space. Our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi-infinite linear matrix inequality or 'robust positive semi-definiteness' of symmetric matrices. In fact, our main result for 3-dimensional cases has been proved by finding an explicit solution range to some semi-infinite linear matrix inequalities.Comment: 24 page

Gravity-mediated holography in fluid dynamics

For any spherically symmetric black hole spacetime with an ideal fluid source, we establish a dual fluid system on a hypersurface near the black hole horizon. The dual fluid is incompressible and obeys Navier-Stokes equation subject to some external force. The force term in the fluid equation consists in two parts, one comes from the curvature of the hypersurface, the other comes from the stress-energy of the bulk fluid.Comment: 12 pages. v2: various corrections. v3: Minor corrections, version to appear in Nucl. Phys.