Alina Stancu

Two Volume Product Inequalities and Their Applications

Abstract. Let K ⊂ R be a convex body of class C with everywhere positive Gauss curvature. We show that there exists a positive number δ(K) such that for any δ ∈ (0, δ(K)) we have Vol(Kδ) · Vol((Kδ) ∗) ≥ Vol(K) · Vol(K∗) ≥ Vol(K) · Vol((K)), where Kδ , K δ and K∗ stand for the convex floating body, the illumination body, and the polar of K , respectively. We derive a few consequences of these inequalities.

Some Affine Invariants Revisited

We present several sharp inequalities for the SL(n) invariant Ω 2, n (K) introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant Ω K defined for convex bodies K whose centroid is at the origin. We offer two alternative definitions for Ω K when K ∈ C + 2. The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized p-affine surface areas of the convex body.

Analysis and geometry of metric measure spaces lecture notes of the 50th Séminaire de Mathématiques Supérieures (SMS), Montréal, 2011

This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential equations. The summer school was designed to lead young scientists to the research frontier concerning the analysis and geometry of metric measure spaces, by exposing them to a series of minicourses featuring leading researchers who highlighted both the state-of-the-art and some of the exciting challenges which remain. This volume attempts to capture the excitement of the summer school itself, presenting the reader with glimpses into this active area of research and its connections with other branches of contemporary mathematics.

Discrete Centro-Affine Curvature for Convex Polygons

We propose a notion of centro-affine curvature for planar, convex polygons which serves to define a non-trivial affine length, and p-affine length respectively, for polygons. These concepts of affine length are shown to be similar to their counterparts defined for smooth convex bodies in that they satisfy analogous affine isoperimetric type inequalities.