scholarly journals A Blichfeldt-type inequality for centrally symmetric convex bodies

2012 ◽  
Vol 170 (3-4) ◽  
pp. 371-379 ◽  
Author(s):  
Matthias Henze
Keyword(s):  
Convex Bodies ◽  
Type Inequality ◽  
Symmetric Convex ◽  
Centrally Symmetric

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

2018 ◽  
Vol 148 (5) ◽  
pp. 913-937
Author(s):  
Dorin Bucur ◽  
Ilaria Fragalà
Keyword(s):  
Convex Body ◽  
Polar Body ◽  
Convex Bodies ◽  
Type Inequality ◽  
Symmetric Convex ◽  
Cheeger Constant ◽  
Regular Triangle ◽  
Centrally Symmetric ◽  
Affine Image ◽  
Krahn Inequality

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

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

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 Self-packing of centrally symmetric convex bodies in ℝ2

1992 ◽  
Vol 8 (2) ◽  
pp. 171-189 ◽  
Author(s):  
P. G. Doyle ◽  
J. C. Lagarias ◽  
D. Randall
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

Centrally symmetric convex bodies

Mathematika ◽  
1984 ◽  
Vol 31 (2) ◽  
pp. 305-322 ◽  
Author(s):  
Paul R. Goodey
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Centrally Symmetric

Contributions to a General Theory of View-Obstruction Problems

1993 ◽  
Vol 45 (3) ◽  
pp. 517-536 ◽  
Author(s):  
V. C. Dumir ◽  
R. J. Hans-Gill ◽  
J. B. Wilker
Keyword(s):  
Critical Parameter ◽  
Original Problem ◽  
Complete Solution ◽  
Convex Bodies ◽  
Positive Cone ◽  
Symmetric Convex ◽  
Reflection Symmetry ◽  
Centrally Symmetric ◽  
Minimal Blocking ◽  
Lower Dimensional

AbstractIn the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

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