scholarly journals Convexity of Bounded Chebyshev Sets in Finite-dimensional Asymmetrically Normed Spaces

2014 ◽  
Vol 14 (4) ◽  
pp. 489-497 ◽  
Author(s):  
A. R. Alimov ◽  
Keyword(s):  
Normed Spaces ◽  
Finite Dimensional ◽  
Chebyshev Sets

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

INTERSECTIONS OF BALLS AND SETS OF CONSTANT WIDTH IN FINITE‐DIMENSIONAL NORMED SPACES

Mathematika ◽  
2013 ◽  
Vol 59 (2) ◽  
pp. 477-492 ◽  
Author(s):  
Horst Martini ◽  
Christian Richter ◽  
Margarita Spirova
Keyword(s):  
Constant Width ◽  
Normed Spaces ◽  
Finite Dimensional

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.

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