We also suppose that A and B are nonempty subsets of a metric space ( X, d ) and use the following notations: d ( A, B ) : = inf { d ( x, y ) : x ∈ A and y ∈ B }, A 0 : = { x ∈ A : d ( x, y ) = d ( A, B ) for some y ∈ B }, B 0 : = { y ∈ B : d ( x, y ) = d ( A, B ) for some x ∈ A }. We observe that if A ∩ B ≠ ∅, then A 0 and B 0 are nonempty.
We also suppose that is a multivalued map such that, for any, is a proper, closed, and convex cone in with,, are vector-valued maps, and, are multivalued maps.
Also suppose that is closed.
We also suppose that a dark appearance and a high gloss correlate with small roughness.
One would also suppose that a large snowflake would have more air resistance than a smaller one.
Also suppose that (a) either F is G b -continuous, or (b) ( X, G ) is regular. .
Also, suppose that (a) T is continuous, or (b) X is regular. .
Also, suppose that (a) F is continuous, or (b) X is regular. .
As customary, we also suppose that a secondary data set,, is available, that each of such snapshots does not contain any useful target echo and exhibits the same covariance matrix as the primary data (homogeneous environment).
Suppose that there exists θ ∈ Θ such that for any sequence { x n } in [ 0, ∞ ) with x n ⟶ t > 0, and for all x, y ∈ X with G x ⪰ G y, d ( T x, T y ) ≤ d ( G x, G y ) − θ ( d ( G x, G y ) ). (3.21). Also, suppose that (a) T is continuous, or (b) X is regular. . T is continuous, or. X is regular.
