A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations.

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is Φp,q(A1, A2, . . . , Am) = (

We compare two approaches to Ricci curvature on nonsmooth spaces in the case of the discrete hypercube {0, 1}N . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm, and Villani could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic nature, includ...

It is shown that if X is a random variable whose density satisfies a Poincaré inequality, and Y is an independent copy of X, then the entropy of (X + Y )/ √ 2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of th...

and that the same holds if any n— 1 of the signs are replaced by strict inequality. Those unimodular lattices, such as Z , which have only the origin in common with the open cube shall be called critical, as shall the corresponding matrices. Minkowski conjectured, and Hajos [ l ] proved in 1938, that a critical lattice must contain one of the points (5»i, • • • , 5,-»), i = 1, • • • , n. If A i...

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube {0, 1} . While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement convexity property of Lott, Sturm and the second author could not be fully implemented. Yet along the way we get new results of a combinatorial and probabilistic natu...

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...

and prove that it is jointly concave for 0 < p ≤ 1 and convex for p = 2. We then derive from this a Minkowski type inequality for operators on a tensor product of three Hilbert spaces, and show how this implies the strong subadditivity of quantum mechanical entropy. For p > 2, Φp is neither convex nor concave. We conjecture that Φp is convex for 1 < p < 2, but our methods do not show this. ∗ Wo...

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...