
Journal of Convex Analysis 26 (2019), No. 3, 877885 Copyright Heldermann Verlag 2019 Extreme Contractions on FiniteDimensional Polygonal Banach Spaces Debmalya Sain Dept. of Mathematics, Indian Institute of Science, Bengaluru 560012, Karnataka, India saindebmalya@gmail.com Anubhab Ray Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India anubhab.jumath@gmail.com Kallol Paul Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India kalloldada@gmail.com [Abstractpdf] We explore extreme contractions on finitedimensional 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 $T \in L(X,Y)$ 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 $x_1, x_2, \ldots, x_n$ of $B_X$ and does not attain norm at any other extreme point of $B_X$, then each $Tx_i$ is an extreme point of $ B_Y.$ We completely characterize extreme contractions between a finitedimensional polygonal Banach space and a strictly convex normed linear space. We introduce LP 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 finitedimensional polygonal Banach space $Y$, the pair $(X,Y)$ does not have LP property. Finally, we obtain a characterization of Hilbert spaces among strictly convex Banach spaces in terms of LP property. Keywords: Extreme contractions, polygonal Banach spaces, strict convexity, Hilbert spaces. MSC: 46B20; 47L05 [ Fulltextpdf (104 KB)] for subscribers only. 