scholarly journals Illuminating Spindle Convex Bodies and Minimizing the Volume of Spherical Sets of Constant Width

2011 ◽  
Vol 47 (2) ◽  
pp. 275-287 ◽  
Author(s):  
Károly Bezdek
Keyword(s):  
Constant Width ◽  
Convex Bodies

SUCCESSIVE RADII OF FAMILIES OF CONVEX BODIES

2014 ◽  
Vol 91 (2) ◽  
pp. 331-344 ◽  
Author(s):  
BERNARDO GONZÁLEZ ◽  
MARÍA A. HERNÁNDEZ CIFRE ◽  
AICKE HINRICHS
Keyword(s):  
Constant Width ◽  
Convex Bodies ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Kolmogorov Numbers

AbstractWe study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the $p$-norms, for which we show that the precise value of the outer (inner) radii when $p\geq 2$ ($1\leq p\leq 2$), as well as bounds in the contrary case $1\leq p\leq 2$ ($p\geq 2$), can be obtained as consequences of known results on Gelfand and Kolmogorov numbers of identity operators between finite-dimensional normed spaces. We also prove properties that successive radii satisfy when we restrict to the families of the constant width sets and the $p$-tangential bodies.

scholarly journals THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

2018 ◽  
Vol 99 (1) ◽  
pp. 130-136
Author(s):  
VITOR BALESTRO ◽  
HORST MARTINI
Keyword(s):  
Differential Geometry ◽  
Upper Bound ◽  
Constant Width ◽  
Convex Bodies ◽  
Euclidean Case ◽  
Planar Convex ◽  
Arbitrary Norms ◽  
Bodies Of Constant Width ◽  
Curves Of Constant Width ◽  
Normed Planes

We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.

scholarly journals On convex bodies of constant width

2006 ◽  
Vol 153 (11) ◽  
pp. 1699-1704 ◽  
Author(s):  
L.E. Bazylevych ◽  
M.M. Zarichnyi
Keyword(s):  
Constant Width ◽  
Convex Bodies ◽  
Bodies Of Constant Width

Convex bodies forming pairs op constant width

1984 ◽  
Vol 22 (2) ◽  
pp. 101-107 ◽  
Author(s):  
Hiroshi Maehara
Keyword(s):  
Constant Width ◽  
Convex Bodies

scholarly journals On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width

2009 ◽  
Vol 52 (3) ◽  
pp. 342-348 ◽  
Author(s):  
K. Bezdek ◽  
Gy. Kiss
Keyword(s):  
Constant Width ◽  
Convex Bodies ◽  
Smooth Convex ◽  
X Ray ◽  
Bodies Of Constant Width

AbstractThe X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

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