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Author: search.filters.author.Pranayanuntana, PoramateSubject: search.filters.subject.Aleksandrov inequality

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    A proof of Sn-i (a11,..., ann) ≥ Sn-i (λ1,..., λn), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [aij]nxn, using mixed determinants
    (2007-01-01)
    Pranayanuntana, Poramate
    By using mixed determinant of special form D (A, n - i; I, i) for positive definite symmetric matrix A, in particular the operator concavity property of the map f : A → D<sup>1/(n-i)</sup> (A, n - i; I, i)I together with unital positive linear map Φ : A → A ο I the Hadamard product of A with the identity matrix I, inequalities of the form S<inf>n-i</inf>(a<inf>11</inf>,..., a<inf>nn</inf>) ≥ S<inf>n-i</inf>(λ<inf>1</inf>,..., λ<inf>n</inf>), 0 ≤ i ≤ n - 1 for an n × n positive definite symmetric matrix A = [a<inf>ij</inf>] with S<inf>k</inf> : IR<sup>n</sup> → IR, k = 1, 2,...,n defined by S<inf>k</inf>(χ) := ∑<inf>1</inf>≤i<inf>1</inf><i<inf>2</inf><...< i<inf>k</inf>≤n χ<inf>i1</inf> χ<inf>i2</inf> · χ<inf>ik</inf>, and called the elementary symmetric polynomials and λ<inf>i</inf>, i = 1, 2,...,n the eigenvalues of A, are derived. This result was first proved using Schur-concavity property of the elementary symmetric function S<inf>k</inf> (χ) together with the fact that for any positive definite symmetric matrix A, the vector of its diagonal entries is majorized by the vector of its eigenvalues. Hence, this suggests relationship among mixed determinant, majorization and Schur-concavity.
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    New matrix inequalities for Firey's extension of Minkowski and Brunn-Minkowski inequalities
    (2006-07-01)
    Pranayanuntana, Poramate
    ;
    Gordon, John
    The Brunn-Minkowski theory is a central part of convex geometry. At its foundation lies the Minkowski addition of convex bodies which led to the definition of mixed volume of convex bodies and to various notions and inequalities in convex geometry. Various matrix analogs of these notions and inequalities have been well known for over a century. We present a few new analogs. The major theorems presented here are the matrix analogs of Firey's Extension of Minkowski inequality and Firey's Extension of Brunn-Minkowski inequality.