Recorded 10 February 2022. Eugenia Saorin-Gomez of the Universität Bremen presents "Inner parallel bodies & the Isoperimetric Quotient" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: The so-called Minkowski difference of convex bodies (compact and convex subsets of R n) can be seen as the subtraction counterpart of the Minkowski or vectorial addition (of convex bodies in R n). ... The monotonicity of the isoperimetric quotient of the family of inner (and outer) parallel bodies of a convex body has received some new attention recently, as it happens, for example, to be connected to the Eikonal abrasion model. We will analyze several aspects of the family of inner parallel bodies of a convex body, in particular, several results about parallel bodies within the realm of the Brunn-Minkowski Theory, based on the concavity of the full system of parallel bodies. Our aim is to prove that the isoperimetric quotient is decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. The result will be obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case.