ISSN:
1572-9281
Keywords:
Banach spaces
;
ideal norms
;
Lp spaces
;
volume ratios
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We extend classical volume formulas for ellipsoids and zonoids to p-sums of segments $${vol}\left( {\sum\limits_{i=1}^m { \oplus_p } [ -x_i ,x_i ]} \right)^{1/n} \sim_{c_p} n^{ - \frac{1}{{p'}}} \left( {\sum\limits_{card(I) = n} {|\det (x_i)_i |^p}} \right)^{\frac{1}{{pn}}}$$ where x1,...,xm are m vectors in $$\mathbb{R}^n ,\frac{1}{p} + \frac{1}{{p\prime }} = 1$$ . According to the definition of Firey, the Minkowski p-sum of segments is given by $$\sum\limits_{i = 1}^m { \oplus _p [ - x_{i,} x_i ]} = \left\{ {\sum\limits_{i = 1}^m {\alpha _i } x_i \left| {\left( {\sum\limits_{i = 1}^m {|\alpha _i |^{p^\prime } } } \right)} \right.^{\frac{1}{{p^\prime }}} \leqslant 1} \right\}.$$ We describe related geometric properties of the Lewis maps associated to classical operator norms.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009731300757
Permalink
