scholarly journals Computational complexity of inner and outerj-radii of polytopes in finite-dimensional normed spaces

1993 ◽  
Vol 59 (1-3) ◽  
pp. 163-213 ◽  
Author(s):  
Peter Gritzmann ◽  
Victor Klee
Keyword(s):  
Computational Complexity ◽  
Normed Spaces ◽  
Finite Dimensional

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

scholarly journals Computational Complexity of Topological Invariants

2014 ◽  
Vol 58 (1) ◽  
pp. 27-32
Author(s):  
Manuel Amann
Keyword(s):  
Computational Complexity ◽  
Decision Problem ◽  
Topological Invariants ◽  
Np Hard ◽  
Connected Space ◽  
Finite Dimensional ◽  
Lusternik Schnirelmann ◽  
Simply Connected

AbstractWe answer the following question posed by Lechuga: given a simply connected spaceXwith bothH*(X; ℚ) and π*(X) ⊗ ℚ being finite dimensional, what is the computational complexity of an algorithm computing the cup length and the rational Lusternik—Schnirelmann category ofX?Basically, by a reduction from the decision problem of whether a given graph isk-colourable fork≥ 3, we show that even stricter versions of the problems above are NP-hard.

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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