Reverse isoperimetric inequality under curvature constraints

SEMINARIO DE GEOMETRÍA

Conferenciante: Kostya Drach (Universidad de Barcelona)

Abstract: What is the smallest volume a convex body \(K\) in \(\mathbb R^n\) can have for a given surface area? This question is in the reverse direction to the classical isoperimetric problem and, as such, has an obvious answer: the infimum of possible volumes is zero. One way to make this question highly non-trivial is to assume that \(K\) is uniformly convex in the following sense. We say that \(K\) is \(\lambda\)-convex if the principal curvatures at every point of its boundary are bounded below by a given constant \(\lambda>0\) (considered in the barrier sense if the boundary is not smooth). By compactness, any smooth strictly convex body in \(\mathbb R^n\) is \(\lambda\)-convex for some \(\lambda>0\). Another example of a \(\lambda\)-convex body is a finite intersection of balls of radius \(1/\lambda\) (sometimes referred to as ball-polyhedra). Until recently, the reverse isoperimetric problem for \(\lambda\)-convex bodies was solved only in dimension 2. In a recent joint work with Kateryna Tatarko, we resolved the problem also in \(\mathbb R^3\). We showed that the lens, i.e., the intersection of two balls of radius \(1/\lambda\), has the smallest volume among all \(\lambda\)-convex bodies of a given surface area. For \(n>3\), the question is still widely open. I will outline the proof of our result and put it in a more general context of reversing classical inequalities under curvature constraints in various ambient spaces.