Approximation of convex bodies by simplices

Novotný, Pavel

doi:10.1007/bf01263651

Cited by 3 publications

(1 citation statement)

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“…3 in . Novotný Novotný (1994) proved that for every C ∈ C 3 we have δ(C, S) ≤ 13 3 . For every C ∈ M n we have δ(C, S) = n (see Grünbaum 1963).…”

Section: Every Simplex Inscribed In D Which Is Not Of the Form Cmentioning

confidence: 99%

Approximation of convex bodies by inscribed simplices of maximum volume

Lassak

2011

Beitr Algebra Geom

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The Banach-Mazur distance between an arbitrary convex body and a simplex in Euclidean n-space E n is at most n +2. We obtain this estimate as an immediate consequence of our theorem which says that for an arbitrary convex body C in E n and for any simplex S of maximum volume contained in C the homothetical copy of S with ratio n + 2 and center in the barycenter of S contains C. In general, this ratio cannot be improved, as it follows from the example of any double-cone.Denote by C n the family of all convex bodies and by M n the family of all centrally symmetric convex bodies in Euclidean n-dimensional space E n .Finding simplices of large volume in convex bodies has a long history. See, for instance, the papers by Blaschke (1917) on the maximum area of triangles in planar convex bodies, by MacKinney McKinney (1974) on simplices of maximum volume in centrally symmetric convex bodies, and the survey article Hudelson et al. (1996) by Hudelson, Klee and Larman on simplices of large volume in cubes.Let A be an (n − 1)-dimensional convex body and let I = ab be a perpendicular to the hyperplane containing A non-degenerated segment centered at a point of A. We call the convex hull conv(A ∪ I ) the double pyramid with base A and apices a and b. If B is a ball of E n − 1 in place of A and if the centers of B and I coincide, the double-pyramid is called a double-cone.

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“…3 in . Novotný Novotný (1994) proved that for every C ∈ C 3 we have δ(C, S) ≤ 13 3 . For every C ∈ M n we have δ(C, S) = n (see Grünbaum 1963).…”

Section: Every Simplex Inscribed In D Which Is Not Of the Form Cmentioning

confidence: 99%

Approximation of convex bodies by inscribed simplices of maximum volume

Lassak

2011

Beitr Algebra Geom

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On the Banach–Mazur Distance in Small Dimensions

Kobos,

Varivoda

2024

Discrete Comput Geom

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We establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in $$\mathbb {R}^3$$ R 3 and $$\mathbb {R}^4$$ R 4 . In dimension three this distance is equal to $$\frac{9}{5}$$ 9 5 , and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

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Approximation of convex sets by polytopes

Bronstein

2008

J Math Sci

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Approximation of convex bodies by simplices

Cited by 3 publications

References 5 publications

Approximation of convex bodies by inscribed simplices of maximum volume

Approximation of convex bodies by inscribed simplices of maximum volume

On the Banach–Mazur Distance in Small Dimensions

Approximation of convex sets by polytopes

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