Given convex bodies K1, . . . ,Kd in R and numbers α1, . . . , αd ∈ [0,1], we give a sufficient condition for existence and uniqueness of an (oriented) halfspace H with Vol(H ∩Ki) = αi · VolKi for every i. The result is extended from convex bodies to measures.
This seemingly innocuous question, considered two decades ago by Bourgain [3, 4], has not been answered yet. We refer the reader to, e.g., [2], [18] or [14] for partial results, history and for additional literature regarding this hyperplane conjecture. In particular, there are l
We discuss valuations on convex sets of oriented hyperplanes in R. For d = 2, we prove an analogue of Hadwiger’s characterization theorem for continuous, rigid motion invariant valuations.
We show that the equivalence of two notions of sufficiency is related to the separation of two orthogonal, strongly convex sets of probability measures by a universally measurable set. Consider a measurable space (äf, sé), and let M(%?) be the set of all probability measures on (
There is an error in the proof of the subadditivity property of IG function on credal sets in the paper "A Non-specificity Measure for Convex Sets of Probability Distributions". The proof is based on the inclusion C C CH(Cx x Cy) which is not always true, being C a credal set on