A Hadwiger-Type Theorem for the Special Unitary Group

Abstract

Abstract. The dimension of the space of SU (n) and translation invariant continuous valuations on C n , n ≥ 2 is computed. For even n, this dimension equals (n 2 + 3n + 10)/2; for odd n it equals (n 2 + 3n + 6)/2. An explicit geometric basis of this space is constructed. The kinematic formulas for SU (n) are obtained as corollaries.

Hermitian integral geometry

“…However this is not true of the unrestricted pairing-it is shown in [11] that if n is odd then the index of the restriction of the pairing to Val SU (n) (C n ) is 1.…”

Integral geometry under G2 and Spin(7)

“…The case G = SU (n) was studied in [10]; again there are no odd invariant valuations. Finally, the case G = G 2 is treated in Theorem 2.2.…”

“…We will need some results from [13] and [10]. Let V be a hermitian vector space of complex dimension n and W ∈ Gr k (V ).…”

“…Using the results of [13] and [10], one computes that Setting η := 5η − μ 4 one gets η · μ 1 = 0 and η · η = 7! 3 μ 8 .…”

“…Problems (i)-(iii) for G = SU(n) were recently solved in [10] and we will use these results in an essential way in the present work.…”