## Journal of Differential Geometry

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

### Structure of the unitary valuation algebra

#### Abstract

S. Alesker has shown that if G is a compact subgroup of O(n) acting
transitively on the unit sphere S^{n-1}, then the vector space Val^{G}
of continuous, translation-invariant, G-invariant convex valuations on R^{n} 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 Val^{U(n)} is isomorphic to R[s, t]/(f_{n+1}, f_{n+2}),
where s, t have degrees 2 and 1 respectively, and the polynomial f_{i} 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

**Permanent link to this document**

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