A bound for the perimeter of inner parallel bodies

• Main Author: Larson, Simon
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: Elsevier 2016
• Subjects: Convex geometry; Perimeter; Inner parallel sets; Ari; laptev
• Online Access: https://doi.org/10.1016/j.jfa.2016.02.022

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body Ω. The bound depends only on the perimeter and inradius r of the original body and states that |∂Ω_t| ≥ (1−t/r)^(n−1)₊|∂Ω|. In particular the bound is independent of any regularity properties of ∂Ω. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes. © 2016 Elsevier Inc. Received 9 November 2015, Accepted 23 February 2016, Available online 3 March 2016. A great deal of gratitude is owed Ari Laptev for many valuable discussions and for proposing the study of the problem at hand. The author would also like to thank Bo Berndtsson for introducing him to convex geometry, and Anders Lundman, Aron Wennman and Eric Larsson for fruitful discussions. Finally we thank the referee for helpful suggestions and comments. The author is supported by Swedish Research Council grant no. 2012-3864. Submitted - 1508.06414.pdf