Continuity properties of selectors and Michael's theorem.

On Lipschitz selections of affine-set valued mappings

“…[PY1], [PY2], [P]). If X is finite-dimensional the classical Steiner point [St] of a convex body provides a suitable selector.…”

“…It is well known that there is a natural connection between extension and selection problems (Whitney's extension problem gives an example of such a connection; see also [PY1]). In particular, every extension problem can be formulated as a selection problem (see [Mi]).…”

“…this definition does not depend on a finite-dimensional subspace L containing C and a Euclidean basis in L. This well-known fact, see, e.g. [PY1], easily follows from the following basic property of the Steiner point: let K(E n ) be the family of all convex compact subsets of an n-dimensional Euclidean vector space equipped with the Hausdorff metric. Let ϕ : K(E n ) → E n be a continuous additive (with respect to the Minkowski addition) map which commutes with the affine isometries of E n .…”

Centres of Convex Sets inLpMetrics

“…Three kinds of centres are widely known and have numerous applications; these are the baricentre, the Steiner point (see: [6,10,11], and especially [12] which is an excellent monograph on this subject) and the Chebyshev centre. We define the baricentre of A # K n by the formula…”

Existence of weak solutions to stochastic evolution inclusions

“…Remark 4.1 It is well known that, if L is infinite dimensional, there exists no Lipschitz selection of K c (L): see [49, Theorem 4], where the reasoning given for the set of convex bounded sets also applies to the set of convex compact sets, and see also [50].…”