Definitions
X
X is a topological vector space.

MINIMAL PAIR
(A,B) is a MINIMAL PAIR if A and B are nonempty bounded convex sets in X and (A,B) is minimal element in quatient class ([A,B], [less than or equal to])

[A,B]
[A,B]={ (C,D)[is an element of] B 2(X) | (A,B) [are equivalent] (C,D) }

(A,B)[less than or equal to](C,D)
(A,B)less than or equal to(C,D) or (A,B) is smaller than (C,D) if (A,B)[are equivalent] (C,D), A[subset of] C and B[subset of] D.

B(X)
B(X) is a family of all nonempty bounded convex subsets of X.

(A,B)[are equivalent](C,D)
Pairs (A,B) and (C,D) are equivalent, if (A,B), (C,D) [is an element of a set] B2(X) and A [Minkowski sum]D = B[Minkowski sum]C

AMinkowski sumB
A[Minkowski sum]B =, closure of A + B

A+B
A+B ={ a + b | a [is an element of]A, b [is an element of] B }- Minkowski sum of A and B

MINIMAL FRACTION
The pair a/b [is an element of a set]S 2 is called MINIMAL FRACTION, if for any fraction c/d such that c/d [are equivalent] a/b and c/d [less then or equal to] a/b it follows that a = c and b = d.

EQUIVALENT FRACTIONS
We say that a/b and c/d are EQUIVALENT FRACTIONS and write a/b [are equivalent] c/d if ad = bc.