I know the following result:
Suppose C and D are nonempty disjoint convex sets. The hyperplane {x | a^T x = b} separates these sets provided a=d−c, b= (∥d∥^2 −∥c∥^2)/2, where c and d are objects lying C and D, respectively, that minimizes the distance.
I wonder if is there another result that ensures a better separation than this?
Thank you!
Hi,
If you search with google on the string "separating hyperplane theorem border" it leads you to a paper on the topic; Theorems 6 and 12 are quite good for your purposes.
Thank Peter, for your comment. Your assumptions are okay: is the T is meaning the transpose of the vector a (what is not difficult to guess, because an hyperplane passing across the origin is the orthogonal complement of some vector, say a, and the others are translations of these ones); and c and d are those vectors in C and D, respectively, that minimizes the distances between these sets.
I would appreciate any further insight!
Thank you, Peter. You are right, perhaps I assumed a lot of things when I wrote the question. By better I am meaning greater margin (i.e., the least distance of d(H,D) and d(H,C), where H denotes the given hyperplane).
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