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Convexity Conditions of Kantorovich Function and Related Semi-infinite Linear Matrix Inequalities
The Kantorovich function , where 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 and thus the convexity of the function with two
variables can be completely characterized by the condition number. The upper
bound `' 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 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.
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