Münster 2026, ca. 70 S. 17 cm x 24 cm
978-3-95987-389-5 Print 19,90 €
978-3-95987-390-1 E-Book 18,90 €
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https://doi.org/10.37626/GA9783959873901.0
Abstract
This monograph develops a matrix-theoretic analog of the classical theory of quermassintegrals from convex geometry, transferring fundamental results — most notably the Minkowski and Brunn–Minkowski inequalities — from the setting of convex bodies to the space of positive definite symmetric matrices. Building on the framework of mixed determinants, mixed matrices, and Blaschke addition introduced in the dissertation work of Pranayanuntana, the author establishes a chain of inequalities and structural results for these matrix functionals.
The exposition begins with foundational inequalities for mixed determinants, including a matrix Minkowski inequality and a matrix Brunn–Minkowski theorem, together with a uniqueness theorem characterizing maps that intertwine with Blaschke summation. Elementary symmetric polynomials are then connected to mixed determinants, yielding inequalities relating the diagonal entries of a positive definite symmetric matrix to its eigenvalues — including a generalization of Hadamard’s inequality — and a recursive formula for a matrix projection operator analogous to Lutwak’s projection operator on convex bodies.
The central contribution is the development of mixed quermassintegrals and their Lp extensions, defined throughₚ directional derivatives with respect to Minkowski and Lp sums of matrices. The principal theorem establishes a Brunn–Minkowski inequality for Lp sums of positive definite symmetric matrices, with equality characterized by scalar proportionality. From this, a corresponding Lp Minkowski inequality for mixed quermassintegrals is derived, shown to be equivalent to the Brunn–Minkowski form, and several uniqueness and equality-case theorems for these functionals are obtained. Four appendices supply the supporting analytic machinery — variational (Rayleigh–Ritz and Courant–Fischer) characterizations of eigenvalues, Weyl-type perturbation results, properties of the Löwner partial order, parallel sums, and integral representations for operator monotone and operator convex functions — making the treatment largely self-contained.