A characterization of affinely regular polygons

Lángi, Zsolt

doi:10.1007/s00010-018-0541-z

Cited by 2 publications

(2 citation statements)

References 15 publications

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“…. , q m of a polygon Q satisfy the condition that q i+2 − q i−1 = τ (q i+1 − q i ) for some τ > 0 independent of i, then Q is an affinely regular m-gon (see [5,6,8], or in a more general form, [18]). This implies that in this case P is affinely regular.…”

Section: Motivated By the Results Of Martini Inmentioning

confidence: 99%

On the convex hull and homothetic convex hull functions of a convex body

Horváth¹,

Lángi²

2020

Preprint

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The aim of this note is to investigate the properties of the convex-hull and the homothetic convex-hull functions of a convex body K in Euclidean n-space, defined as the volume of the union of K and one of its translates, and the volume of K and a translate of a homothetic copy of K, respectively, as functions of the translation vector.In particular, we prove that the convex-hull function of the body K do not determine K. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Castro about the homothetic convex-hull function, and prove a homothetic variant of the translative constant volume property conjecture for 3-dimensional convex polyhedra.

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Section: Motivated By the Results Of Martini Inmentioning

confidence: 99%

On the convex hull and homothetic convex hull functions of a convex body

Horváth¹,

Lángi²

2020

Preprint

Self Cite

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show abstract

“…. , q m of a polygon Q satisfy the condition that q i+2 − q i−1 = τ (q i+1 − q i ) for some τ > 0 independent of i, then Q is an affinely regular m-gon (see [30], [31], [32], or in a more general form, [33]). This implies that in this case P is affinely regular.…”

Section: Convex Bodies and G K λ (T) Depends Only On The L-norm ||T|| L Of Tmentioning

confidence: 99%

On the convex hull and homothetic convex hull functions of a convex body

Horváth

Lángi

2022

Geom Dedicata

Self Cite

View full text Add to dashboard Cite

The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body K in Euclidean n-space, defined as the volume of the union of K and one of its translates, and the volume of K and a translate of a homothetic copy of K, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body K does not determine K. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jerónimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for 3-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies.

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A characterization of affinely regular polygons

Cited by 2 publications

References 15 publications

On the convex hull and homothetic convex hull functions of a convex body

On the convex hull and homothetic convex hull functions of a convex body

On the convex hull and homothetic convex hull functions of a convex body

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