Invariant Valuations on Star-Shaped Sets

Valuations on Polytopes Containing the Origin in Their Interiors

“…In the dual Brunn-Minkowski theory, Klain gave a classification of continuous, rotation invariant valuations on star-shaped sets [11,12]. In particular, for continuous valuations invariant with respect to the special linear group SLðdÞ; 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.…”

Valuations on Polytopes Containing the Origin in Their Interiors

“…In the dual Brunn-Minkowski theory, Klain gave a classification of continuous, rotation invariant valuations on star-shaped sets [11], [12]. 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.…”

Affine invariant points

“…This is due to, on the one hand, rapid progress in core affine areas directly linked to the very structure of convex bodies, like the L p Brunn Minkowski theory (e.g., [2,5,7,8,11], [18]- [26], [30,35,36,38,39]) and the theory of valuations [10], [12], [13], [15]- [17], [32,33]. On the other hand, even questions that had been considered Euclidean in nature turned out to be affine problems-among them the famous Busemann-Petty Problem (finally laid to rest in [3,6,41,42]).…”