Hölder’s inequality states that ‖x‖p ‖y‖q − 〈x, y〉 ≥ 0 for any (x, y) ∈ Lp(Ω) × Lq(Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q−2: ‖x‖p ‖y‖q − 〈x, y〉 ≥ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] ≥ 0 if p ∈ (1, 2], 0 ≤ ‖x‖p ‖y‖q − 〈x, y〉 ≤ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] if p ≥ 2. A similar result holds for complex valued fun...

In this paper we extend some theorems published lately on the relationship between convexity/concavity, and subadditivity/superadditivity. We also generalize inequalities of compound functions that refine Minkowski inequality.

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

The classical Hlawka inequality possesses deep connections with zonotopes and zonoids in convex geometry, has been related to Minkowski space. We introduce Type-1 Type-2 quantities, establish a Hlawka-type relation between them, which connects vast number of strikingly different variants the inequalities, such as Serre's reverse future cone space, for subadditive function on abelian group by Re...

In this paper, we present the Bennett-type generalization bounds of the learning process for i.i.d. samples, and then show that the generalization bounds have a faster rate of convergence than the traditional results. In particular, we first develop two types of Bennett-type deviation inequality for the i.i.d. learning process: one provides the generalization bounds based on the uniform entropy...

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the Lp analogue of the classical BusemannPetty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corr...

Digital subscriber lines (DSLs) are fundamentally limited by crosstalk. The case where all crosstalk is from the same type of DSL has been studied over the years and accurate models have been standardized. However, crosstalk from multiple different types of DSLs is a relatively new area of study and models of summing mixed crosstalk have only recently been postulated. As more and more types of ...