We prove an analogue of the classical Steiner formula for the Lp affine surface area of a Minkowski outer parallel body for any real parameters p. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature. They have the intrinsic volumes of the classical Brunn Minkowski theory and the dual quermassintegrals of the dual Brunn Minkowski theory as well as special cases.Properties of these new quantities are investigated, a connection to information theory among them. A Steiner formula for the s-th mixed Lp affine surface area of a Minkowski outer parallel body for any real parameters p and s is also given. *

Let A = (a ij ) be a square n × n matrix with i.i.d. zero mean and unit variance entries. It was shown by Rudelson and Vershynin in 2008 that the upper bound for the smallest singular value s n (A) is of order n − 1 2 with probability close to one under the additional assumption that the entries of A satisfy Ea 4 11 < ∞. We remove the assumption on the fourth moment and show the upper bound assuming only Ea 2 11 = 1.

Inspired by an Lp Steiner formula for the Lp affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, Lp-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the Lp affine surface areas and the mixed Lp affine surface areas. We investigate the properties of the Lp-Steiner quermassintegrals. In particular, we show that they are rotation and reflection invariant valuations on the set of convex bodies with a certain degree of homogeneity. Such valuations seem new and not have been observed before.

We consider the class of λ-concave bodies in R n+1 ; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius 1/λ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius 1/λ (a sausage body) is a unique volume minimizer among all λ-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper.

Let Γ be an N × n random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope Γ * B N 1 in R n (the absolute convex hull of rows of Γ). In particular, we show thatwhere b depends only on parameters in small ball inequality. This extends results of [18] and recent results of [17]. This inclusion is equivalent to so-called ℓ 1 -quotient property and plays an important role in compressed sensing (see [17] and references therein).