Gaussian variables, polynomials and permanents
It is proved that for \(n\) given unit vectors \(({\mathbf a}_{j})^{n}_{j=1}\) in a complex Hilbert space there exists such a unit vector \(\mathbf x\) that the inequality \( |\langle {\mathbf x}, {\mathbf a}_{1} \rangle\dots\langle {\mathbf x},{\mathbf a}_{n} \rangle |\geq n^{-n/2}\) holds. This result is derived on the basis of similar bounds for norms in real Hilbert and Banach spaces and on the basis of properties of complex Gaussian variables and permanents of a positive semidefinite Hermitian \(n \times n\) matrix.
scientific article; zbMATH DE number 1371027
scientific article