ON THE EXTREMAL SEMI-LIPSCHITZ FUNCTIONS

. The extremal elements of the unit balls of Banach spaces play an important role in the study of the geometry of the space as well in various applications. For Banach spaces of Lipschitz real functions the extremal elements of the unit ball are investigates in numerous papers (S. Cobzas 1989, J. D. Farmer 1994, N. V. Rao and A. C. Roy 1970, Roy 1968 and in the references therein). In this note we shall present a procedure to obtain extremal elements of the unit ball of the quasi-normed semilinear space of real-valued semi-Lipschitz functions deﬁned on a quasi-metric space.


INTRODUCTION
Let X be a nonvoid set.A function d : X → [0, ∞] is called a quasi -metric if it satisfies the conditions: (i) d (x, y) = d (y, x) = 0 ⇐⇒ x = y (ii) d (x, y) ≤ d (x, z) + d (z, y) or (i ) d (x, y) = 0 ⇐⇒ x = y and (ii), for all x, y, z ∈ X.The pair (X, d) is called a quasi -metric space.
Remark that d is not a symmetric function, i.e., it is possible that d (x, y) = d (y, x) for x, y ∈ X.
A function f : X → R, defined on a quasi -metric space (X, d) is called semi-Lipschitz if there exists K ≥ 0 such that (1) f (x) − f (y) ≤ K • d (x, y) , for all x, y ∈ X.
(see [12]), where The set SLipX defined in (2) is exactly the set of all semi-Lipschitz functions on (X, d), and f X defined by ( 3) is the least semi-Lipschitz constant for f , i.e. ( 4) and every K ≥ 0, for which the inequality (1) holds, satisfies K ≥ f X (see [9] and [12]).For x 0 ∈ X be fixed, denote by ( 5) the set of all real-valued semi-Lipschitz functions defined on the quasi-metric space X which vanish at the fixed point x 0 ∈ X.
If X is a linear space then a functional • X : X → [0, ∞) satisfying the axioms of a quasi-norm is called an asymmetric norm on X (see [4]).
It is immediate that the functional defined by ( 3) is a quasi-norm on SLip 0 X, i.e. the pair (SLip 0 X, • X ) is a quasi-normed semilinear space.
If Y ⊂ X and x 0 ∈ Y then one considers the semi-Lipschitz functions on Y which vanish at x 0 and the quasi-normed semilinear space (S Lip 0 Y, • Y ), where • Y is defined like in (3) with Y instead of X.
The following extension theorem for semi-Lipschitz functions is similar to Mc Shane's [6] extension theorem for Lipschitz functions.
Theorem 1. [9].Let (X, d) be a quasi-metric space, x 0 ∈ X fixed and Y ⊂ X such that x 0 ∈ Y .Then every function f ∈ SLip 0 Y admits at least one extension in SLip 0 X, i.e. there exists H ∈ SLip 0 X such that (6) H| Y = f and Denote by the nonvoid set of all extensions of f ∈ SLip 0 Y which preserve the quasi-norm of f .We have shown in [9] that the functions ( 8) be the unit ball of the quasi-normed semilinear space (SLip 0 Y, • Y ) and let B X be the corresponding unit ball of (SLip 0 Y, hold for all x ∈ X, where the functions F and G are defined by (8) and (9), respectively; (defined by ( 8) and ( 9)) are extremal elements of B X .
Proof.a) Let F 1 , F 2 ∈ E Y (f ) and α ∈ (0, 1) .We have On the other hand Taking the infimum with respect to y ∈ Y we find H (x) ≤ F (x) , for all x ∈ X.

Also, we have
Taking the supremum with respect to y ∈ Y we get We remark that F, G defined by ( 8), ( 9)are extremal elements of On the other hand x ∈ X and λ ∈ (0, 1) it follows 2 • .The assertion c) from Theorem 2 gives us a way to obtain extremal elements of B X , namely as the extensions (8) and ( 9) of extremal elements of B Y .