instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
On geometric inequalities and their functional counterparts
A convex body $K\subset\mathbb{R}^n$ is a convex and compact set with non-empty interior. For each $K$ and real $p\geq 1$, we can define $\Gamma_pK$ the $p$-centroid body of $K$. The $L_p$ Busemann-Petty centroid inequality states that $V(\Gamma_pK) \geq V(K)$, with equality if only if $K$ is an ellipsoid centered at the origin. Also, for each $K$ we can define the petty projection body $\Pi K$ of $K$ and consider $\Pi^{\circ}K$ the polar body of $\Pi K$. The Petty Projection inequality states that $V(K)^{n-1}V(\Pi^{\circ}K) \leq \left(\frac{\omega_n}{\omega_{n-1}}\right)^n$, with equality if and only if $K$ is a ellipsoid. 
For a geometric inequality, its functional form (or extension) is an integral inequality which reduces to the former by an appropriate choice of the functions involved. In this talk, we will present the joint work with J. Haddad and C. H. Jim\'enez about functional forms for $L_p$ Busemann-Petty centroid inequality, and the joint work with B. Gonz\'alez Merino and R. Villa about the Petty Projection inequality and related inequalities for log-concave functions.