# Characterization of rotation equivariant additive mappings Franz Schuster Technical University Vienna This work is supported by the Austrian Science Fund.

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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".

Definition:  : K n  K n is a rotation equivariant additive map if (i)  is continuous (ii)  is equivariant with respect to SO(n)  ( K ) = (K )   SO(n) (iii)  is additive  (K L) =  (K )  (L)  K, L  K n either Minkowski or Blaschke addition

Minkowski addition: Blaschke addition: S n – 1 (K # L,. ) = S n – 1 (K,. ) + S n – 1 (L,. )  (K,)(K,) S n – 1 (K,  ) = vol n – 1 (  (K,  )) K S n – 1 o - h(K,- u) u K h(K,u)h(K,u) h(K + L,. ) = h(K,. ) + h(L,. )

Examples: s: K n  IR n is a Minkowski endomorphism Steiner point Projection operator  : K n  K n is a Blaschke Minkowski homomorphism   (K # L) =  K +  L

Theorem [Schneider,74]:  : K 2  K 2 is a Minkowski endomorphism if and only if weakly positive there is a weakly positive measure µ  M(S 1 ) such that weakly positive A measure is called weakly positive if it is nonnegative up to addition of a measure with density u x. u. For n = 2 Minkowski endomorphisms are weakly monotone weakly monotone:  K  L and s(K ) = s(L) = o   K   L

Theorem [Kiderlen,99]: zonal A measure is called zonal if it is SO(n – 1) invariant. h(  K,. ) = h(K,. ) * µ. weakly monotone weakly positive  : K n  K n is a weakly monotone Minkowski endomorphism if and only if there is a weakly positive zonal zonal measure µ  M(S n – 1 ) such that  : K n  K n is a Blaschke endomorphism if and only if weakly positive zonal there is a weakly positive zonal measure µ  M(S n – 1 ) such that S n – 1 (  K,. ) = S n – 1 (K,. ) * µ.

  is a 'sum' of symmetric bodies of revolution Theorem [F.S.,04]: weakly positive zonal If  : K n  K n is a Blaschke Minkowski homomorphism there is a weakly positive zonal function g  C(S n – 1 ) such that symmetric  : K n  K n is a symmetric Blaschke Minkowski symmetric body homomorphism if and only if there is a symmetric body of revolution of revolution L  K n such that h(  K,. ) = S n – 1 (K,. ) * g. h(  K,. ) = S n – 1 (K,. ) * h(L,. ).

Multiplier transformations:  Injectivity of  is equivalent to c k  0 Example [Goodey, Weil, 92]: If  : K n  K n is a Blaschke Minkowski homomorphism there is a sequence of real numbers c k such that  k : M(S n – 1 )  H k n is the orthogonal projection on the space H k n of spherical harmonics of degree k  k h(  K,. ) = c k  k S n – 1 (K,. )  K  K n.

Petty‘s conjectured projection inequality: "=" only if  K is a ball Steiner formula for  : i = n – 2 "=" only for ellipsoids ? Lutwak‘s extension of Petty's conjecture: ? i = 0, …, n – 2

Steiner formula for B-M-homomorphisms: Theorem [F.S.,05]: "=" only if  n – 2 K is a ball "=" only for balls i = 0, …, n – 3 Conjecture: ?

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