Continuous Rotation Invariant Valuations on Convex Sets

Valuations on Polytopes Containing the Origin in Their Interiors

“…In particular, for continuous valuations invariant with respect to the special linear group SL(d), i.e., the group of linear transformations with determinant 1, he showed that the only examples are linear combinations of the Euler characteristic and volume. The difficult problem of classifying continuous, rotation invariant valuations on the space of convex bodies was solved by Alesker [2], [3]. In his proof, he approximated continuous, rotation invariant valuations by polynomial valuations and then classified the latter, making use of representations of the orthogonal group.…”

“…In his proof, he approximated continuous, rotation invariant valuations by polynomial valuations and then classified the latter, making use of representations of the orthogonal group. As an application of his results, he obtained a classification of tensor or polynomial valued, continuous, rigid motion covariant valuations on the space of convex bodies [4].…”

Rotation Invariant Valuations

“…In the seminal paper [1] by Alesker, characterization theorems for rotation invariant continuous polynomial valuations are derived. A valuation ϕ :…”

“…In [1], a characterization theorem for continuous polynomial rotation invariant valuations is derived, involving the family of valuations given by …”

Valuations on Convex Sets of Oriented Hyperplanes

“…The surveys [25] and [24] show how this celebrated result is embedded in the theory of valuations on convex bodies. In recent years this theory has been enriched by various classification and characterization results for valuations under different assumptions, which were all inspired by Hadwiger's theorem; see [1]- [7], [12]- [23], and [26].…”