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Fusion of multiple estimates by covariance intersection: Why and how it is suboptimal

Ajgl J., Straka O.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 3

521--530

The fusion under unknown correlations tunes a combination of local estimates in such a way that upper bounds of the admissible mean square error matrices are optimised. Based on the recently discovered relation between the admissible matrices and Minkowski sums of ellipsoids, the optimality of existing algorithms is analysed. Simple examples are used to indicate the reasons for the suboptimality of the covariance intersection fusion of multiple estimates. Further, an extension of the existing family of upper bounds is proposed, which makes it possible to get closer to the optimum, and a general case is discussed. All results are obtained analytically and illustrated graphically.

Strongly monotypic polytopes in R3

Borowska D., Grzybowski J.

Demonstratio Mathematica

2008

Vol. 41, nr 1

165-169

McMullen et al. [3] characterized all three-dimensional monotypic polytopes by presenting four types of such polytopes. They stated that three of these types are strongly monotypic polytopes. We examine which monotypic polytopes of the fourth type are strongly monotypic. This way we complete the description of all strongly monotypic polytopes in R3.

Relative distance of boundary points of a convex body and touching by homothetical copies

Lángi Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 4

439-444

The relative distance of points p and g of a convex body C is the ratio of the length of the segment pq to the half of the length of a longest chord of C parallel to pq. In this paper we find a connection between pairwise relative distances of k points in the boundary of a convex body and the rafio of k homothetical copies of a convex body touching it.

On common summands and anti-summands of compact convex sets

Grzybowski J., Urbański A., Wiernowolski M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 1

69-76

In this paper we examine families of common summands and anti-summands of compact convex sets. In particular, we prove the equivalence of minimality problems for common anti-summands and minimal pairs of compact convex sets. We also give partial answers to the question whether minimal anti-summands of polytopes are polytopes as well.