Random subspaces of proportional dimension of finite dimensional normed spaces: Approach through the isoperimetric inequality

1985 ◽  
pp. 106-115 ◽  
Author(s):  
V. D. Milman
Keyword(s):  
Isoperimetric Inequality ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Random Subspaces

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.

scholarly journals Finite dimensional characteristics related to superreflexivity of Banach spaces

2004 ◽  
Vol 69 (2) ◽  
pp. 289-295 ◽  
Author(s):  
M. I. Ostrovskii
Keyword(s):  
Banach Spaces ◽  
Normed Space ◽  
Local Theory ◽  
Normed Spaces ◽  
Finite Sets ◽  
Large Cardinality ◽  
Finite Dimensional ◽  
Finite Set

One of the important problems of the local theory of Banach Spaces can be stated in the following way. We consider a condition on finite sets in normed spaces that makes sense for a finite set any cardinality. Suppose that the condition is such that to each set satisfying it there corresponds a constant describing “how well” the set satisfies the condition.The problem is:Suppose that a normed space X has a set of large cardinality satisfying the condition with “poor” constant. Does there exist in X a set of smaller cardinality satisfying the condition with a better constant?In the paper this problem is studied for conditions associated with one of R.C. James's characterisations of superreflexivity.

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