Since K is a convex set
, the line segment [[[omega].sub.k], [[omega].sub.k]] connecting [[omega].sub.k] and [[omega].sub.k] lies entirely inside K, and ([[omega].sub.k] - [[omega].sub.k]) is a feasible direction.
Each linear term in (7) corresponds to the projection onto the convex set
in the signal or Fourier domains in the Gerchberg algorithm.
A tropical convex set
in [T.sup.n] is a subset closed under [direct sum] and scaling by elements of T, that is, a T-subsemimodule of [T.sup.n].
If for every x, y [member of] I we need that x should be an end point of the path, then I reduces to a convex set
where [THETA] [subset] [R.sup.nxn] is a nonempty closed convex set
, [xi] is a random matrix whose probability distribution P is supported onset [OMEGA] [subset] [R.sup.nxn], and F is a smooth convex stochastic function on [THETA] x [OMEGA].
Let C and S* be the subclasses of S which are convex (f(D) is a convex set
) and starlike (f(D) is a starlike set with respect to the origin) respectively.
For example, it is well-known that every ||*||-closed (originally closed, strongly closed) convex set
is also [T.sub.w]-closed (weakly closed).
Recall that a convex space X is a nonempty convex set
with any topology that induces the Euclidean topology on the convex hulls of its finite subsets.
We may do so by imposing the fact that the volume also belongs to some convex set
(a set is convex if, for any two volumes in this set, [x.sub.1] and [x.sub.2], the linear combination (1 - [lambda])[x.sub.1] + [lambda][x.sub.2], with 0 [less than or equal to] [lambda] [less than or equal to] 1, also belongs to that set; the set of all nonnegative volumes is convex as well as the set of all volumes defined within a mask, the set of all volumes with a given symmetry, the set of all volumes bandlimited to a given frequency, etc.
We say that x [member of] A is an extremal point for the convex set
A c N if x does not belong to the interior of some geodesic segment with the ends in A.
Since [OMEGA] is a convex set
and [bar.t], [t.sup.*] [member of] [OMEGA], we have