| In this paper we study the following problems: given a finite number of nonemptyclosed subsets of a normed space, find a ball with the smallest radius that encloses all of thesets, and find a ball with the smallest radius that intersects all of the sets. These problems canbe viewed as generalized versions of the smallest enclosing circle problem introduced in the19th century by Sylvester which asks for the circle of smallest radius enclosing a given setof finite points in the plane. We will focus on the sufcient conditions for the existence anduniqueness of an optimal solution for each problem, while the study of optimality conditionsand numerical implementation will be addressed in our next project. After that, we initiatethe study of minimal time functions generated by unbounded dynamics and discuss theirapplications to further extensions of the smallest enclosing ball problem.In chapter2,lemma1Suppose that Q is a nonempty bounded set of X. Then the minimal functionCF(x; Q)=inf{t≥0: Q x+tF} has the following representation:CF(x; Q)=sup{ρF(w x): w∈}.Moreover, if F is the closed unit ball of X, then CF(x; Q)=sup{w x: w∈}.theorem1Suppose one of the following holdsμ(i) The constraint is a compact set.(ii) X is a reflexive Banach space and the constraint set is a weakly closed set.Then the smallest enclosing ball problem has a solution. That means there exit x∈andr≥0such that i x+rF i=1,..., n, and anyx∈and t≥0such that i x+rFfori=1,..., n§one has r≤t.we can prove the existence of the smallest enclosing ball problem. Then with the theorem2the sufcient condition can be given.theorem2, Suppose that X=Rn, F is the Euclidean closed unit ball of X, and theconstraint is a nonempty closed convex subset of X.Then our problem has a unique solu-tion.Moreover, we give two examples to illustrate the need for assumptions in the above theoremthat provides the uniqueness of the the solution to the smallest enclosing ball problem.In chapter3, we havetheorem3, Assume that one of the following statements holds:(i) X is finite dimensional, and one of the i,i=1,..., n and is bounded.(ii) X is a reflexive Banach space, all of the sets i,i=1,..., n and are weakly closed,andat least one of them is bounded.Then the smallest intersecting ball problem has a solution, In this case there exists r≥0andx∈such that(x+rF)∩i i=1,..., n.and for any x∈and t≥0with (x+rF)∩i for all i=1,...,n,one has r≤t.theorem4let X be a Hilbert space,and let F be the closed,unit ball of X. Supposethat is a nonempty,closed,convex set, i(i=1,..., n) are strictly convex, and at least oneof the sets among iand is bounded. Suppose further that∩in=1[i∩]=.our optimization problem has a unique solution.Then we give two examples to illustrate the need for assumptions in the above theorem thatprovides the uniqueness of the the solution to the smallest intersecting ball problem.In chapter4, we introduce some methods for the minimal time functions and the small-est intersecting ball problem with unbounded dynamics,and some better properties briefly. lemma2For all x1,x2∈F, we haveF∞(x1)=F∞(x2),that is, the asymptotic cone does not depend on x∈F.Thus,we will denote the asymptoticcone as F∞.theorem5Assume that one of the following statements holds:(i) X be a reflexive Banach space,Q be a bounded,weakly closed set, and F contains theorigin.(ii) Q be a compact set and F contains the origin.ThenTF(x; Q)=0i f and only i f x∈Q F∞.At last we prove the function TF(x; Q) is lower semi-continuous. |
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