We prove the smallest possible norm of a linear perturbation making a closed convex process nonsurjective is the inverse of the norm of the inverse process. This generalizes the fundamental property of the condition number of a linear map. We then apply this result to strengthen a theorem of Renegar measuring the size of perturbation necessary to make an inequality system inconsistent.

The classical Trudinger-Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H 0 (Ω) (with Ω ⊂ R a bounded domain), the integral ∫ Ω e 2 dx is uniformly bounded by a constant depending only on Ω. If the volume |Ω| becomes unbounded then this bound tends to infinity, and hence the Trudinger-Moser inequality is not available for such domains (and...

We prove the Gagliardo–Nirenberg-type multiplicative interpolation inequality ‖v‖L1(Rn) ≤ C‖v‖ 1/2 Lip′(Rn)‖v‖ 1/2 BV(Rn) ∀v ∈ Lip ′(Rn) ∩ BV(R), where C is a positive constant, independent of v. Here ‖·‖Lip′(Rn) is the norm of the dual to the Lipschitz space Lip 0(R) := C 0,1 0 (Rn) = C0,1(Rn) ∩C0(R) and ‖ · ‖BV(Rn) signifies the norm in the space BV(Rn) consisting of functions of bounded vari...

* Correspondence: limin-zou@163. com School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404100, People’s Republic of China Abstract We obtain an improved Heinz inequality for scalars and we use it to establish an inequality for the Hilbert-Schmidt norm of matrices, which is a refinement of a result due to Kittaneh. Mathematical Subject Classification 2010: 26D07;...

Several new trace norm inequalities are established for 2n × 2n block matrices, in the special case where the four n × n blocks are diagonal. Some of the inequalities are non-commutative analogs of Hanner's inequality , others describe the behavior of the trace norm under reordering of diagonal entries of the blocks.

We investigate a convex function ψp,q,λ = max{ψp, λψq}, (1 ≤ q < p ≤ ∞), and its corresponding absolute normalized norm ‖.‖ψp,q,λ . We determine a dual norm and use it for getting refinements of the classical Hölder inequality. Also, we consider a related concave function φp,q,λ = min{ψp, λψq}, (0 < p < q ≤ 1).

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.

We conjecture the following so-called norm compression inequality for 2×N partitioned block matrices and the Schatten p-norms: for p ≥ 2,