Uniqueness Results for Extremal Quardircs

By a famous theorem of affine convex geometry a full-dimensional, bounded and compact subset F of Rd can be enclosed by a unique ellipsoid of minimal volume (the Löwner ellipsoid to F). Furthermore, the convex hull of F contains a unique ellipsoid of maximal volume (the John ellipsoid to F). Both, minimal and maximal volume ellipsoids have numerous applications in diverse fields of pure and applied mathematics, among them convex geometry, robotics, computer graphics, statistics, computer aided design, and computational geometry. Existence of extremal ellipsoids is a consequence of compactness of F. Uniqueness of extremal volume ellipsoids has been proved by Fritz John in 1948 and Danzer, Laugwitz and Lenz in 1957. Further contributions on uniqueness of minimal ellipsoids are due to Firey (1964) and Schröcker (2008), and in 2007 Schröcker proved a uniqueness result for minimal enclosing hyperbolas of line-sets. In the proposed project we will derive uniqueness results of extremal quadrics, thereby extending existing results in different ways. In particular we plan to show uniqueness of maximal enclosed ellipsoids with respect to size functions different from the volume, uniqueness of extremal hyperboloids to sets of subspaces (for example extremal hyperboloids of one sheet to sets of lines in R3 or extremal hyperboloids of two sheets to sets of planes in R3 ) and uniqueness results for extremal quadrics in non-Euclidean geometries. Extremal quadrics in this sense have potential applications in geometric tolerancing, computer aided design, surveying or image processing. While uniqueness results are our primal interest we also consider, if appropriate, characterizations of extremal quadrics, their geometric properties, results on their approximation quality and algorithms for their efficient computation. The techniques used for the investigation of extremal quadrics come from analysis, analytic geometry, optimisation theory, and convex geometry.

By a famous theorem of affine convex geometry a full-dimensional, bounded and compact subset F of Rd can be enclosed by a unique ellipsoid of minimal volume (the Löwner ellipsoid to F). Furthermore, the convex hull of F contains a unique ellipsoid of maximal volume (the John ellipsoid to F). Both, minimal and maximal volume ellipsoids have numerous applications in diverse fields of pure and applied mathematics, among them convex geometry, robotics, computer graphics, statistics, computer aided design, and computational geometry. Existence of extremal ellipsoids is a consequence of compactness of F. Uniqueness of extremal volume ellipsoids has been proved by Fritz John in 1948 and Danzer, Laugwitz and Lenz in 1957. Further contributions on uniqueness of minimal ellipsoids are due to Firey (1964) and Schröcker (2008), and in 2007 Schröcker proved a uniqueness result for minimal enclosing hyperbolas of line-sets. In the proposed project we will derive uniqueness results of extremal quadrics, thereby extending existing results in different ways. In particular we plan to show uniqueness of maximal enclosed ellipsoids with respect to size functions different from the volume, uniqueness of extremal hyperboloids to sets of subspaces (for example extremal hyperboloids of one sheet to sets of lines in R3 or extremal hyperboloids of two sheets to sets of planes in R3 ) and uniqueness results for extremal quadrics in non-Euclidean geometries. Extremal quadrics in this sense have potential applications in geometric tolerancing, computer aided design, surveying or image processing. While uniqueness results are our primal interest we also consider, if appropriate, characterizations of extremal quadrics, their geometric properties, results on their approximation quality and algorithms for their efficient computation. The techniques used for the investigation of extremal quadrics come from analysis, analytic geometry, optimisation theory, and convex geometry.