David Alonso-Gutiérrez

Factoring Sobolev inequalities through classes of functions

We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, [21], [3], while the second one relies on tools from Convex Geometry, [32], [16]. In this paper we prove a (sharp) connection between them.

Mean width of random perturbations of random polytopes

Abstract We prove some “high probability” results on the expected value of the mean width for random perturbations of random polytopes. The random perturbations are considered for Gaussian random vectors and uniform distributions on ℓ p N

A REMARK ON THE ISOTROPY CONSTANT OF POLYTOPES

\ell_p^N

John’s Ellipsoid and the Integral Ratio of a Log-Concave Function

-balls and the unit sphere.

Estimates for the Integrals of Powered i -th Mean Curvatures

It is known that the isotropy constant of any symmetric polytope with 2N vertices is bounded by C log N. We give a different proof of this result, which shows that the same estimate is true when the polytope is non-symmetric with N vertices. We also make a remark on how an estimate of the isotropy constant of a symmetric polytope with 2N facets of the order of √log N/n, which can be easily deduced from known results, is also true for non-symmetric polytopes with N facets.

A Reverse Rogers–Shephard Inequality for Log-Concave Functions

We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.

Large deviations for high-dimensional random projections ofℓpn-balls

We show (upper and lower) estimates for the integrals of powered i-th mean curvatures, i = 1, …, n − 1, of compact and convex hypersurfaces, in terms of the quermasintegrals of the corresponding \(C^2_+\)-convex bodies. These bounds are obtained as consequences of a most general result for functions defined on a general probability space. Moreover, similar estimates for the integrals of powers of the elementary symmetric functions of the radii of curvature of \(C^2_+\)-convex bodies are proved. This probabilistic result will also allow to get new inequalities for the dual quermasintegrals of starshaped sets, via further estimates for the integrals of the composition of a convex/concave function with the (powered) radial function.

Convex inequalities, isoperimetry and spectral gap

We will prove a reverse Rogers–Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of

On the isotropy constant of projections of polytopes

\ell _p