An origin-symmetric convex body $K$ in $\mathbb{R}^n$ is said to be in the John position if the maximal volume ellipsoid contained in it is the Euclidean ball.

In 1961, Grunbaum asked whether the centroid $c(K)$ of a convex body $K$ is the centroid of at least $n + 1$ different $(n − 1)$-dimensional sections of $K$ through $c(K)$. A few years later, Loewner asked to find the minimum number of hyperplane sections of $K$ passing through $c(K)$ whose centroid is the same as $c(K)$.

The Brascamp–Lieb inequalities are an important family of inequalities in analysis that subsumes several inequalities significant in their own right, including Hölder’s inequality, Young’s inequality, and the Loomis–Whitney inequality. Several variants and extensions of these inequalities have been developed, some of which have proved to be very useful in contemporary harmonic analysis.

Paraproducts are one of the essential tools of harmonic analysis, used to decompose a product of two functions. Motivated by a similar question in complex analysis, Pott, Reguera, Sawyer and Wick studied the composition of "paraproduct-type operators", to which classical paraproducts belong. Their goal was to find joint conditions on the symbol functions for the composition to be bounded. They classified many paraproduct-type compositions. One of the operators to remain unclassified was the composition of two classical dyadic paraproducts.

We discuss a kind of weak-type inequality for the Hardy-Littlewood maximal operator that was first studied by Muckenhoupt and Wheeden. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior; in particular, as shown by Muckenhoupt and Wheeden, the class of weights characterizing such inequalities is strictly larger than $A_p$.

We generalize the classical Vieta formulas that express the coefficients of a polynomial in terms of all the roots. In particular, we focus on the case when just some of the roots are known. Our formulas are established by exploiting    some properties of the complete homogeneous symmetric polynomials. 
Further, we provide  new identities for these polynomials when the variables are in a geometric progression and give an application to cyclotomic polynomials. This is joint work with  A. Echezabal and M. Laporta

We discuss two problems in stochastic geometry:

1. Sylvester's problem is a classical question that asks for the probability that d+2 random points in R^d are in convex position, that is, that none of them is in the convex hull of the others. We compute this probability when the points are the steps of a random walk. Remarkably, the probability depends only on dimension and is independent of the increment distribution of the walk, provided that it satisfies a mild nondegeneracy condition.

Davies's efficient covering theorem states that we can cover any measurable set in the plane by lines without increasing the total measure. This result has a dual formulation, known as Falconer's digital sundial theorem, which states that we can construct a set in the plane to have any desired projections, up to null sets. The argument relies on a Venetian blind construction, a classical method in geometric measure theory.

We establish a multiversion of real and complex Gaussian hypercontractivity. More precisely, our result generalizes Nelson’s hypercontractivity in the real setting and the works of Beckner, Weissler, Janson, and Epperson in the complex setting to several functions. The proof relies on heat semigroup methods, where we construct an interpolation map that connects the inequality at the endpoints. As a consequence, we derive sharp multiversion of the Hausdorff-Young inequality and the log-Sobolev inequality. This is joint work with Paata Ivanisvili.

The radial mean body of parameter $p>-1$ of a convex body $K \subseteq \mathbb R^n$ is a radial set $R_p K$ that was introduced by Gardner and Zhang in 1998.

They proved that if $p \geq 0$, then $R_p K$ is convex, and conjectured that this holds also for $p \in (-1, 0)$.

We prove that if $K \subseteq \mathbb R^2$ is a convex body in the plane, then $R_p K$ is convex for every $p > (-1,0)$.