Volume of Convex Hull of Two Bodies and Related Problems

Horváth, Ákos G.

doi:10.1007/978-3-319-78434-2_11

Cited by 4 publications

(4 citation statements)

References 32 publications

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“…He determined those supporting hyperplanes H of the regular simplex S for which the volume of S ′ H is maximal or minimal, respectively. Similar interesting questions can be found in the paper [4] on the volume of the union of two convex body and also in the survey paper [5].…”

Section: Introductionsupporting

confidence: 63%

An extremal problem of regular simplices: the five-dimensional case

Horváth

2019

J. Geom.

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The new result of this paper connected with the following problem: Consider a supporting hyperplane of a regular simplex and its reflected image at this hyperplane. When will be the volume of the convex hull of these two simplices maximal? We prove that in the case when the dimension is less or equal to 4, the maximal volume achieves in that case when the hyperplane goes through on a vertex and orthogonal to the height of the simplex at this vertex. More interesting that in the higher dimensional cases this position is not optimal. We also determine the optimal position of hyperplane in the 5-dimensional case. This corrects an erroneous statement in my paper [3].√ 2 e i , where {e 0 , e 1 , . . . , e n } is an orthonormed basis of an (n + 1)dimensional Euclidean space. Set s i := 1 √ 2 (e i − e 0 ) for i = 0, . . . , n the system of the 2010 Mathematics Subject Classification. 52A40, 52A38, 26B15, 52B11.

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Section: Introductionsupporting

confidence: 63%

An extremal problem of regular simplices: the five-dimensional case

Horváth

2019

J. Geom.

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“…Horváth and Lángi [26] extended the methods in [4] and determined the maximum volume polytope in S d−1 with d + 2 vertices, d ≥ 2, and also with d + 3 vertices when d is odd. The general question of determining the maximal volume polytope inscribed in S d−1 with v ≥ d + 1 of vertices is asked in, e.g., the problem books [10,13] and in [25,19]. To the best of our knowledge, this problem is unsolved for arbitrary d and v ≥ d + 4 (the general case is also unknown for surface area).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$

Donahue,

Hoehner,

2020

Preprint

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“…The convex hull function gave rise to a number of interesting problems, many of which are still open; for a collection of such problems see the survey paper [6] and the references therein. Nevertheless, it is an interesting fact that the 'dual' of the covariogram problem, that is, the question whether the convex hull function G K (t) determines the body K or not has been asked only recently by Á. Kurusa in a private communication.…”

Section: Introductionmentioning

confidence: 99%

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

Horváth

Lángi

2022

Geom Dedicata

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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 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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Volume of Convex Hull of Two Bodies and Related Problems

Abstract: In this paper we deal with problems concerning the volume of the convex hull of two "connecting" bodies. After a historical background we collect some results, methods and open problems, respectively.2010 Mathematics Subject Classification. 52B60, 52A40, 52A38.

Cited by 4 publications

References 32 publications

An extremal problem of regular simplices: the five-dimensional case

An extremal problem of regular simplices: the five-dimensional case

The Maximum Surface Area Polyhedron with Five Vertices Inscribed in the Sphere $\mathbb{S}^2$

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

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