Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila, Benedetti & Doker (2010). The coefficients yI of such a Minkowski decompositio...

Using the methods developed by Fewster and colleagues, we derive a quantum inequality for the free massive spin2 Rarita-Schwinger fields in the four dimensional Minkowski spacetime. Our quantum inequality bound for the RaritaSchwinger fields is weaker, by a factor of 2, than that for the spin2 Dirac fields. This fact along with other quantum inequalities obtained by various other authors for th...

In this work we consider robust control of discrete-time linear systems affected by time-varying additive disturbance inputs. We present a linear matrix inequality (LMI) based design technique that takes into account in an explicit manner, by means of a Minkowski function, the shape of the set in which the disturbances are bounded. This technique allows one to obtain tight bounds on the perform...

β1 ψ dx− ab+ α1β1. Young’s inequality [2, 1, 4] states that for every a ∈ [α1, α2] and b ∈ [β1, β2] (2) 0 ≤ F (a, b), where the equality holds iff φ(a) = b (or, equivalently, ψ(b) = a). Among the classical inequalities Young’s inequality is probably the most intuitive. Indeed, its meaning can be easily grasped once the integrals are regarded as areas below and on the left of the graph of φ (see...

Jensen’s inequality is sometimes called the king of inequalities [4] because it implies at once the main part of the other classical inequalities (e.g. those by Hölder, Minkowski, Young, and the AGM inequality, etc.). Therefore, it is worth studying it thoroughly and refine it from different points of view. There are numerous refinements of Jensen’s inequality, see e.g. [3-5] and the references...

The entropy power inequality, which plays a fundamental role in information theory and probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by this connection to Convex Geometry, we survey various recent developments on forward and reverse entropy power inequalities not just for the Shannon-Boltzmann entropy but also more generally for Rényi entropy. In the proce...

We prove a concentration inequality for the `q norm on the `p sphere for p, q > 0. This inequality, which generalizes results of Schechtman and Zinn, is used to study the distance between the cone measure and surface measure on the sphere of `p . In particular, we obtain a significant strengthening of the inequality derived in [NR], and calculate the precise dependence of the constants that app...

Here we collect some notation and basic lemmas used throughout this note. Throughout, for a random variable X, ‖X‖p denotes (E |X|). It is known that ‖ · ‖p is a norm for any p ≥ 1 (Minkowski’s inequality). It is also known ‖X‖p ≤ ‖X‖q whenever p ≤ q. Henceforth, whenever we discuss ‖ · ‖p, we will assume p ≥ 1. Lemma 1 (Khintchine inequality). For any p ≥ 1, x ∈ R, and (σi) independent Rademac...

It is well known that isoperimetric inequalities imply in a very general measuremetric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition...

We study a Riemannian manifold equipped with a density which satisfies the Bakry–Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, origin...