Characterization of rotation equivariant additive mappings

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Characterization of rotation equivariant additive
mappings
Franz Schuster Technical University Vienna
This work is supported by the Austrian Science
Fund within the scope of the project "Affinely
associated bodies" and the European Community
within the scope of the project "Phenomena in
high dimensions".
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Definition
? K n ? K n is a rotation equivariant additive
map if
(i) ? is continuous
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Minkowski addition
Blaschke addition
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Examples
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Theorem Schneider,74
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Theorem Kiderlen,99
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Theorem F.S.,04
If ? K n ? K n is a Blaschke Minkowski
homomorphism there is a weakly positive zonal
function g ? C(S n 1) such that
h(?K, . ) Sn 1(K, . ) g.
? K n ? K n is a symmetric Blaschke
Minkowski homomorphism if and only if there is a
symmetric body of revolution L ? K n such that
h(?K, . ) Sn 1(K, . ) h(L, . ).
? ? is a 'sum' of symmetric bodies of revolution
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Multiplier transformations
? Injectivity of ? is equivalent to ck ? 0
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Pettys conjectured projection inequality
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Steiner formula for B-M-homomorphisms