This monograph develops a matrix-theoretic analog of the classical theory of quermassintegrals from convex geometry, transferring fun-damental results — most notably the Minkowski and Brunn–Minkow-ski 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 intro-duced in the dissertation work of Pranayanuntana, the author estab-lishes a chain of inequalities and structural results for these matrix functionals.
The exposition begins with foundational inequalities for mixed deter-minants, including a matrix Minkowski inequality and a matrix Brunn–Minkowski theorem, together with a uniqueness theorem characteriz-ing maps that intertwine with Blaschke summation. Elementary sym-metric polynomials are then connected to mixed determinants, yield-ing inequalities relating the diagonal entries of a positive definite sym-metric matrix to its eigenvalues — including a generalization of Hada-mard’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 quermassinte-grals and their Lₚ extensions, defined through directional derivatives with respect to Minkowski and Lₚ sums of matrices. The principal the-orem establishes a Brunn–Minkowski inequality for Lₚ sums of posi-tive 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 pertur-bation results, properties of the Löwner partial order, parallel sums, and integral representations for operator monotone and operator con-vex functions — making the treatment largely self-contained.

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