Asymmetry of Convex Polytopes and Vertex Index of Symmetric Convex Bodies

2008 ◽  
Vol 40 (4) ◽  
pp. 528-536 ◽  
Author(s):  
E. D. Gluskin ◽  
A. E. Litvak
Keyword(s):  
Convex Polytopes ◽  
Convex Bodies ◽  
Symmetric Convex ◽  
Vertex Index

Volume ratios for Cartesian products of convex bodies

2021 ◽  
Vol 32 (5) ◽  
pp. 905-916
Author(s):  
A. Khrabrov
Keyword(s):  
Convex Bodies ◽  
Volume Ratio ◽  
Symmetric Convex ◽  
Partial Differential ◽  
Cartesian Products ◽  
Content Type ◽  
Vertex Index

The paper is devoted to the behavior of volume ratios, the modified Banach–Mazur distance, and the vertex index for sums of convex bodies. It is shown that sup d ( A ⊕ K , B ⊕ L ) ≥ sup ∂ ( A ⊕ K , B ⊕ L ) ≥ c ⋅ n 1 − k + k ′ 2 n , \begin{equation*} \sup d (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq \sup \partial (\mathrm {A}\oplus \mathrm {K},\mathrm {B}\oplus \mathrm {L}) \geq c \cdot n^{1-\frac {k+k’}{2n}}, \end{equation*} if K ⊂ R n \mathrm {K}\subset \mathbb {R}^n and L ⊂ R k \mathrm {L}\subset \mathbb {R}^k are convex and symmetric (the supremum is taken over all symmetric convex bodies A ⊂ R n − k \mathrm {A}\subset \mathbb {R}^{n-k} and B ⊂ R n − k ′ ) \mathrm {B}\subset \mathbb {R}^{n-k’}) . Furthermore, some examples are discussed that show that the available extimates of the vertex index in terms of the volume ratio are not sharp.

scholarly journals A measure of axial symmetry of centrally symmetric convex bodies

2010 ◽  
Vol 121 (2) ◽  
pp. 295-306 ◽  
Author(s):  
Marek Lassak ◽  
Monika Nowicka
Keyword(s):  
Axial Symmetry ◽  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

scholarly journals Subspace concentration of dual curvature measures of symmetric convex bodies

2018 ◽  
Vol 109 (3) ◽  
pp. 411-429 ◽  
Author(s):  
Károly J. Böröczky ◽  
Martin Henk ◽  
Hannes Pollehn
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Curvature Measures

scholarly journals Non-symmetric convex polytopes and Gabor orthonormal bases

2018 ◽  
Vol 146 (12) ◽  
pp. 5147-5155
Author(s):  
Randolf Chung ◽  
Chun-kit Lai
Keyword(s):  
Convex Polytopes ◽  
Symmetric Convex ◽  
Orthonormal Bases

A characterization of centrally symmetric convex bodies in En

1981 ◽  
Vol 10 (1-4) ◽  
pp. 161-176 ◽  
Author(s):  
D. G. Larman ◽  
N. K. Tamvakis
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

Some problems concerning centrally symmetric convex bodies

1978 ◽  
Vol 10 (3) ◽  
pp. 454-460
Author(s):  
V. A. Zalgaller ◽  
V. N. Sudakov
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

scholarly journals Centrally symmetric convex bodies and sections having maximal quermassintegrals

2012 ◽  
Vol 49 (2) ◽  
pp. 189-199
Author(s):  
E. Makai ◽  
H. Martini
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
The Body ◽  
Symmetric Convex ◽  
Local Theorem ◽  
Centrally Symmetric ◽  
Euclidean Unit Ball ◽  
Analogous Theorem ◽  
Lower Dimensional ◽  
Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

scholarly journals Random approximation and the vertex index of convex bodies

2016 ◽  
Vol 108 (2) ◽  
pp. 209-221 ◽  
Author(s):  
Silouanos Brazitikos ◽  
Giorgos Chasapis ◽  
Labrini Hioni
Keyword(s):  
Convex Bodies ◽  
Vertex Index
Sign in / Sign up

Export Citation Format

Share Document