The sum of squared logarithms inequality in arbitrary dimensions

  title={The sum of squared logarithms inequality in arbitrary dimensions},
  author={Lev Borisov and Patrizio Neff and Suvrit Sra and Christian Thiel},
We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝn whose elementary symmetric polynomials satisfy ek(x) ≤ ek(y) (for 1 ≤ k < n) and en(x) = en(y) , the inequality ∑i(log xi)2 ≤ ∑i(log yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂn → ℝ with f(z) = ∑i(log zi)2 has nonnegative partial derivatives with respect to the elementary symmetric… 

Logarithmic inequalities under an elementary symmetric polynomial dominance order

We consider a dominance order on positive vectors induced by the elementary symmetric polynomials. Under this dominance order we provide conditions that yield simple proofs of several monotonicity

Inequalities via symmetric polynomial majorization

We consider a partial order on positive vectors induced by elementary symmetric polynomials. As a corollary we obtain a short proof of the SSLI inequality of Neff et al. (2012), which was first

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

We consider the problem to determine the optimal rotations R∈SO(n) which minimize W:SO(n)→R0+,W(R;D)≔sym(RD−1)2for a given diagonal matrix D≔diag(d1,…,dn)∈Rn×n with positive entries di>0 . The

Explicit Global Minimization of the Symmetrized Euclidean Distance by a Characterization of Real Matrices with Symmetric Square

The optimal orthogonal matrices are determined which minimize the symmetrized Euclidean distance W(R,;D) =vert\vert{{sym}(R D - 1) .

New Thoughts in Nonlinear Elasticity Theory via Hencky’s Logarithmic Strain Tensor

We consider the two logarithmic strain measures \(\omega _{\mathrm {iso}}= ||{{\mathrm{dev}}}_n \log U ||\) and \(\omega _{\mathrm {vol}}= |{{\mathrm{tr}}}(\log U) |\), which are isotropic invariants

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field F:=∇φ:Ω→ GL +(n) and the microrotation field R:Ω→ SO (n) , the

SCHRIFTENREIHE DER FAKULTÄT FÜR MATHEMATIK An Elementary Method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations by

This paper presents a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations using a combined norm taking into account both the primal and dual variables.


From a conceptual point of view, two-stage stochastic programs and bilevel problems under stochastic uncertainty are closely related. However, the step from the first to the latter mirrors the step

The modified indeterminate couple stress model: Why Yang et al.'s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless

We show that the reasoning in favor of a symmetric couple stress tensor in Yang et al.'s introduction of the modified couple stress theory contains a gap, but we present a reasonable physical

A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

The unitary polar factor Q = Up in the polar decomposition of Z = Up H is the minimizer over unitary matrices Q for both kLog(QZ)k 2 and its Hermitian part ksym * (Log(QZ))k 2 over both R and C for

Symmetric polynomials in information theory: entropy and subentropy

It is seen that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory.

On the sum of squared logarithms inequality and related inequalities

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors

Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

AbstractLet y1,y2,y3,a1,a2,a3∈(0,∞) be such that y1y2y3=a1a2a3 and y1+y2+y3≥a1+a2+a3,y1y2+y2y3+y1y3≥a1a2+a2a3+a1a3. Then

Towards a geometrical interpretation of quantum-information compression (6 pages)

Let S be the von Neumann entropy of a finite ensemble E of pure quantum states. We show that S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the

On the generalised sum of squared logarithms inequality

Assume n≥2$n\geq2$. Consider the elementary symmetric polynomials ek(y1,y2,…,yn)$e_{k}(y_{1},y_{2},\ldots, y_{n})$ and denote by E0,E1,…,En−1$E_{0},E_{1},\ldots,E_{n-1}$ the elementary symmetric

Riemannian geometry and matrix geometric means

Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices

This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.