Totally invariant set functions of polynomial type

Radial Minkowski additive operators

“…In the 1930s, Blaschke started a systematic investigation, and then Hadwiger obtained the famous Hadwiger's characterization theorem. The Hadwiger's characterization theorem provides the connection between rigid motion invariant set functions and symmetric polynomials (see [4]) for further results and generalizations, see [1], [2], [3], [5], [20], [29]. The following Minkowski endomorphism was introduced by Schneider in [21].…”

Invariant Valuations on Star-Shaped Sets

“…Hadwiger's characterization leads to effortless proofs of numerous results in geometric convexity, including mean projection formulas for convex bodies [13, p. 294] and various kinematic formulas [7,12,14,15]. Hadwiger's theorem also provides a connection between rigid motion invariant set functions and symmetric polynomials [1,7].…”

Star Valuations and Dual Mixed Volumes

“…Hadwiger's characterization leads to effortless proofs of numerous results in integral geometry, including the mean projection formulas for convex bodies [22] and various kinematic formulas [20,24]. This result also provides a connection between rigid motion invariant set functions and symmetric polynomials [2,20]. Hadwiger's theorem is of such fundamental importance that any candidate for a dual theory must possess a dual analogue of this theorem.…”