Decomposition of polytopes using inner parallel bodies

Linke, Eva; Gómez, Eugenia Saorín

doi:10.1007/s00605-014-0629-z

Cited by 3 publications

(3 citation statements)

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“…Both of these properties are demonstrated in Figure 1. Nonetheless, the theory of inner parallel sets is rich and has several beautiful applications in both convex geometry and analysis (see for instance [2,4,10,12,13]). …”

Section: Introductionmentioning

confidence: 99%

A bound for the perimeter of inner parallel bodies

Larson

2016

Journal of Functional Analysis

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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.

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Section: Introductionmentioning

confidence: 99%

A bound for the perimeter of inner parallel bodies

Larson

2016

Journal of Functional Analysis

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“…Inner parallel bodies of convex bodies have been the object of recent studies with different flavors [7,[12][13][14][15]18,20]. Classical references on inner parallel bodies (e.g.…”

Section: Introductionmentioning

confidence: 99%

On the Monotonicity of the Isoperimetric Quotient for Parallel Bodies

Richter

Gómez

2021

J Geom Anal

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The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be 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. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained.

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Section: Introductionmentioning

confidence: 99%

A bound for the perimeter of inner parallel bodies

Larson

2015

Preprint

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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 thatIn 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.

show abstract

Decomposition of polytopes using inner parallel bodies

Cited by 3 publications

References 10 publications

A bound for the perimeter of inner parallel bodies

A bound for the perimeter of inner parallel bodies

On the Monotonicity of the Isoperimetric Quotient for Parallel Bodies

A bound for the perimeter of inner parallel bodies

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