Restricted Invertibility and the Banach-mazur Distance to the Cube

On the Banach-mazur Distance between the Cube and the Crosspolytope

In this note we study the Banach-Mazur distance between the n-dimensional cube and the crosspolytope. Previous work shows that the distance has order √ n, and here we will prove some explicit bounds improving on former results. Even in dimension 3 the exact distance is not known, and based on computational results it is conjectured to be 9 5 . Here we will also present computerbased potential o...

A Remark on the Mahler Conjecture: Local Minimality of the Unit Cube

We prove that the unit cube Bn ∞ is a strict local minimizer for the Mahler volume product voln(K)voln(K ∗) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.

Banach-Mazur Distances and Projections on Random Subgaussian Polytopes

We consider polytopes in Rn that are generated by N vectors in Rn whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random ±1 polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially of a maximal order n. This result is a...

A Survey of Invertibility and Spectrum Preserving Linear Maps

On the relations between the point spectrum of A and invertibility of I + f(A)B

Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I+f(A)B is invertible for every analytic function f on sigma(A). Also we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space ell^2 for solving some PDEs which is extended to general operator space under some ...