It is well known that among all convex bodies in R^n with a given surface area, the Euclidean ball has the largest inradius. We will show that this result can be reversed in the class of convex bodies with curvature at each point of their boundary bounded below by some positive constant λ (λ-convex bodies). In particular, we show that among λ-convex bodies of a given surface area,  the λ-convex lens (the intersection of two balls of radius 1/ λ) minimizes the inradius.

We study extremal properties of spherical random polytopes, the convex
hull of random points chosen from the unit Euclidean sphere in Rn. The extremal
properties of interest are the expected values of the maximum and minimum surface area among facets.
We determine the asymptotic growth in every fixed dimension, up to absolute constants.

Inequalities for Minkowski sums of convex bodies play a central role in convex geometry and additive combinatorics. A particular  example is the Plünnecke–Ruzsa-type inequality, which also has a geometric analogue for convex bodies. In this talk, we begin by introducing such geometric inequalities and highlighting some of the open problems related to them. We then present functional analogues of Plünnecke–Ruzsa-type inequalities using the alpha-sum, a generalized sup-convolution.

I shall present an extension of Webb's simplex slicing (1996) to the analytic framework of sharp L_p bounds for centred log-concave measures on the real line, with a curious phase transition of the extremising distribution. Based on joint work with Melbourne, Roysdon and Tang.
 

Abstract: Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever and however far from the body we want, to illuminate its entire surface. What is the minimum number of such light sources that we would have to use? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes.

In this talk we will discuss old and not so old inequalities on the
volume of the orthogonal projections (sometimes called local
Loomis-Whitney type estimates).  We will explore connections of those
inequalities to inequalities for mixed volumes as well as inequalities
of the Minkwoski sums of convex bodies.
 

Two convex bodies $K$ and $L$ in $\mathbb{R}^n$ are affine-equivalent if there exists an affine transformation $T$ such that $T(K)=L$. To measure how close two convex bodies are when they are not exactly affinely equivalent, one introduces the Banach--Mazur distance. Roughly speaking, this is the smallest factor $R \ge 1$ such that, after an appropriate affine transformation, one body is contained in the other, and the other is contained in its dilation by $R$ (with respect to some center $\xi$).