##
**Continuous rotation invariant valuations on convex sets.**
*(English)*
Zbl 0941.52002

By a classical theorem of Hadwiger, a rigid motion invariant, continuous (real) valuation on the space \({\mathcal K}^d\) of convex bodies in Euclidean \(d\)-space is a finite linear combination of intrinsic volumes.

In the present paper, the author succeeds in establishing an analogous (and deeper) result without the assumption of translation invariance. The first theorem states that every continuous \(SO(d)-\) (resp. \(O(d)-)\) invariant valuation on \({\mathcal K}^d\) can be approximated uniformly on compact subsets of \({\mathcal K}^d\) by continuous polynomial \(SO(d)-\) (resp. \(O(d)-)\) invariant valuations. A valuation \(\varphi\) is called polynomial of degree at most \(\ell\) if \(\varphi(K+x)\) is a polynomial in \(x\) of degree at most \(\ell\) for each \(K\in{\mathcal K}^d\). Examples are the functionals \(\xi_{p,q}\) defined by \[ \xi_{p,q} (K)=\int_{\partial K} \bigl\langle s,n(s)\bigr\rangle^p |s|^{2q} d{\mathcal H}^{d-1}(s) \] \((n(s)\) outer unit normal vector at \(s\in\partial K\), \({\mathcal H}^{d-1}(d-1)\)-dimensional Hausdorff measure) and the coefficients \(\xi^{(j)}_{p,q}\) in the expansion \[ \xi_{p,q} (K+\varepsilon D)= \sum^{p+2q+d}_{j=0} \varepsilon^j \xi_{p,q}^{(j)} (K), \] where \(D\) is the unit ball. The author’s main result says that, if \(d\geq 3\), every \(SO(d)\)-invariant continuous polynomial valuation on \({\mathcal K}^d\) is a finite linear combination of the \(\xi^{(j)}_{p,q}\). A variant of this result holds for \(d=2\). The proof uses several results on valuations and employs representations of the rotation group. The author concludes with applications to integral-geometric formulas, some inequalities for related functionals, and a few open questions.

{Reviewer’s remark. As a remarkable application, the author has recently obtained a classification of the continuous, isometry covariant (tensor or polynomial valued) valuations [Geom. Dedicata 74, No. 3, 241–248 (1999; Zbl 0935.52006)], thus answering a question posed by P. McMullen [in: Second international conference in stochastic geometry, convex bodies and empirical measures, Agrigento, Italy, September 9–14, 1996. Palermo: Circolo Matematico di Palermo. 259–271 (1997; Zbl 0901.52009)]}.

In the present paper, the author succeeds in establishing an analogous (and deeper) result without the assumption of translation invariance. The first theorem states that every continuous \(SO(d)-\) (resp. \(O(d)-)\) invariant valuation on \({\mathcal K}^d\) can be approximated uniformly on compact subsets of \({\mathcal K}^d\) by continuous polynomial \(SO(d)-\) (resp. \(O(d)-)\) invariant valuations. A valuation \(\varphi\) is called polynomial of degree at most \(\ell\) if \(\varphi(K+x)\) is a polynomial in \(x\) of degree at most \(\ell\) for each \(K\in{\mathcal K}^d\). Examples are the functionals \(\xi_{p,q}\) defined by \[ \xi_{p,q} (K)=\int_{\partial K} \bigl\langle s,n(s)\bigr\rangle^p |s|^{2q} d{\mathcal H}^{d-1}(s) \] \((n(s)\) outer unit normal vector at \(s\in\partial K\), \({\mathcal H}^{d-1}(d-1)\)-dimensional Hausdorff measure) and the coefficients \(\xi^{(j)}_{p,q}\) in the expansion \[ \xi_{p,q} (K+\varepsilon D)= \sum^{p+2q+d}_{j=0} \varepsilon^j \xi_{p,q}^{(j)} (K), \] where \(D\) is the unit ball. The author’s main result says that, if \(d\geq 3\), every \(SO(d)\)-invariant continuous polynomial valuation on \({\mathcal K}^d\) is a finite linear combination of the \(\xi^{(j)}_{p,q}\). A variant of this result holds for \(d=2\). The proof uses several results on valuations and employs representations of the rotation group. The author concludes with applications to integral-geometric formulas, some inequalities for related functionals, and a few open questions.

{Reviewer’s remark. As a remarkable application, the author has recently obtained a classification of the continuous, isometry covariant (tensor or polynomial valued) valuations [Geom. Dedicata 74, No. 3, 241–248 (1999; Zbl 0935.52006)], thus answering a question posed by P. McMullen [in: Second international conference in stochastic geometry, convex bodies and empirical measures, Agrigento, Italy, September 9–14, 1996. Palermo: Circolo Matematico di Palermo. 259–271 (1997; Zbl 0901.52009)]}.

Reviewer: R.Schneider (Freiburg i.Br.)

### MSC:

52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |