## Journal of Differential Geometry

### Structure of the unitary valuation algebra

Joseph H.G. Fu

#### Abstract

S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere Sn-1, then the vector space ValG of continuous, translation-invariant, G-invariant convex valuations on Rn has the structure of a finite dimensional graded algebra over R satisfying Poincaré duality. We show that the kinematic formulas for G are determined by the product pairing. Using this result we then show that the algebra ValU(n) is isomorphic to R[s, t]/(fn+1, fn+2), where s, t have degrees 2 and 1 respectively, and the polynomial fi is the degree i term of the power series log(1 + s + t).

#### Article information

Source
J. Differential Geom., Volume 72, Number 3 (2006), 509-533.

Dates
First available in Project Euclid: 28 March 2006

https://projecteuclid.org/euclid.jdg/1143593748

Digital Object Identifier
doi:10.4310/jdg/1143593748

Mathematical Reviews number (MathSciNet)
MR2219942

Zentralblatt MATH identifier
1096.52003

#### Citation

Fu, Joseph H.G. Structure of the unitary valuation algebra. J. Differential Geom. 72 (2006), no. 3, 509--533. doi:10.4310/jdg/1143593748. https://projecteuclid.org/euclid.jdg/1143593748