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  • 1
    Electronic Resource
    Electronic Resource
    Volume Formulas in Lp-Spaces (1997)
    Springer
    Positivity 1 (1997), S. 7-43 
    Details
    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
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  • 2
    Electronic Resource
    Electronic Resource
    Constructing a polytope to approximate a convex body (1995)
    Springer
    Geometriae dedicata 57 (1995), S. 217-222 
    Details
    ISSN: 1572-9168
    Keywords: 52A20 ; 52A25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We develop an algorithm to construct a convex polytopeP withn vertices, contained in an arbitrary convex bodyK inR d , so that the ratio of the volumes |K/P|/|K| is dominated byc ·. d/n 2/(d−1).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01264939
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Majorization of Gaussian processes and geometric applications (1992)
    Springer
    Probability theory and related fields 91 (1992), S. 251-267 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let (x i * ) i=1 n denote the decreasing rearrangement of a sequence of real numbers (x i ) i=1 n . Then for everyi≠j, and every 1≦k≦n, the 2-nd order partial distributional derivatives satisfy the inequality, $$\frac{{\partial ^2 }}{{\partial x_i \partial x_j }}\left( {\sum\limits_{l = 1}^k {x_l^* } } \right) \leqq 0$$ . As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in ℝ n proved by Gromov.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01291425
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  • 4
    Electronic Resource
    Electronic Resource
    Elliptically contoured distributions (1987)
    Springer
    Probability theory and related fields 76 (1987), S. 429-438 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Given two covariance matricesR andS for a given elliptically contoured distribution, we show how simple inequalities between the matrix elements imply thatE R(f)≦E S(f), e.g., whenx=(xi1,i2,...,in) is a multiindex vector and $$f(x) = \mathop {\min }\limits_{i_1 } \mathop {\max }\limits_{i_2 } \mathop {\min }\limits_{i_3 } \max ...x_{i_{1,...,} i_n } ,$$ orf(x) is the indicator function of sets such as $$\mathop \cap \limits_{i_1 } \mathop \cup \limits_{i_2 } \mathop \cap \limits_{i_3 } \cup ...[x_{i_{1,...,} i_n } \mathop〈 \limits_ = \lambda _{i_{1,...,} i_n } ]$$ of which the well known Slepian's inequality (n=1) is a special case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00960067
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    Paper (German National Licenses)
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