A bound for the perimeter of inner parallel bodies

Abstract

Abstract. 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 |∂Ωt| ≥ 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.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

“…It might be worth noting that both bounds in (49) cannot be improved. In the upper bound equality is achieved if Ω is a ball and, more generally, if and only if Ω is a form body (see [22,29]). In the lower bound equality is asymptotically achieved by (0, L) d−1 × (0, 1) in the limit L → ∞.…”

Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions

“…Since every tangential body to a ball is homothetic to its form body [19] (in our case Ω(η) is in fact equal to its form body), the main result in [12] implies…”

“…To prove (12) we recall the definition and some basic properties of mixed volumes [19, p. 275ff]. Let K denote the set of convex bodies in R n with nonempty interior.…”

“…In order to prove (12) we need a bound from below. For the sum in the numerator it suffices to keep the first two terms in the expansion and to use Property (1) of W resulting in n j=0…”

The Isoperimetric Quotient of a Convex Body Decreases Monotonically Under the Eikonal Abrasion Model

“…While our result highlights a fundamental property of the nonlinear PDE (1), our proof will rely entirely on ideas and concepts from convex geometry. In particular, as one can see in Section 2, the examined problem, and also the applied tools, are closely related to those used in [23], where, by an elegant method, the author gave a sharp lower estimate on the surface area of an inner parallel body of a convex body. These bodies also play a central role in the current paper: Equation (1) can be regarded as a map from a convex body to one of its inner parallel bodies where time plays the role of the distance of this body from the original one.…”