Extreme Contractions on Finite-Dimensional Polygonal Banach Spaces

We explore extreme contractions on finite-dimensional polygonal Banach spaces, from the point of view of attainment of norm of a linear operator. We prove that if X is an n-dimensional polygonal Banach space and Y is any normed linear space and T is an element of L(X, Y) is an extreme contraction, then T attains norm at n linearly independent extreme points of B-X. Moreover, if T attains norm at n linearly independent extreme points x(1), x(2), ..., x(n), of B-X and does not attain norm at any other extreme point of B-X, then each T-x(i), is an extreme point of B-Y. We completely characterize extreme contractions between a finite-dimensional polygonal Banach space and a strictly convex normed linear space. We introduce L-P property for a pair of Banach spaces and show that it has natural connections with our present study. We also prove that for any strictly convex Banach space X and any finite-dimensional polygonal Banach space Y, the pair (X, Y) does not have L-P property. Finally, we obtain a characterization of Hilbert spaces among strictly convex Banach spaces in terms of L-P property