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We characterize extreme and smooth points in Lorentz sequence space d(w, 1) and in Marcinkiewicz sequence spaces d∗(w, 1) and d∗(w, 1), which are predual and dual spaces to d(w, 1), respectively. We then apply these characterizations for studying the relationship between the existence and one-complemented subspaces in d(w, 1). We show that a subspace of d(w, 1) is an existence set if and only if it is one-complemented. Marcinkiewicz and Lorentz spaces play an important role in the theory of Banach spaces. They are key objects for instance in the interpolation theory of linear operators. The origins of the Marcinkiewicz spaces go back to the theorem on weak type operators [23, th. 2.b.15], which was originally due to K. Marcinkiewicz in the 1930-ties. The Lorentz spaces introduced by G.G. Lorentz in 1950, have appeared in a natural way as interpolation spaces between suitable Lebesgue spaces by classical result of Lions and Peetre [23, th. 2.g.18]. This theory has been developed very extensively thereafter and along with these investigations, the theory of Lorentz and Marcinkiewicz spaces, including the studies of their geometric structure, have been evolved independently (e.g. [6, 7, 22, 25]). One can observe that these spaces find also applications in other topics of operator theory. It is worth to mention that Marcinkiewicz spaces d∗(w, 1) have emerged recently many times in the context of norm-attaining linear operators. In the papers [1, 9, 14] it was shown among others, by using the space d∗(w, 1) with specific weight, that the subspace of norm attaining operators is not always dense in the space of all bounded operators, contrary to the Bishop-Phelps theorem for linear functionals. For such types of isometric results the knowledge of geometric properties of the ball is of the utmost importance (see e.g. [9], where the characterization of complex convexity of the Lorentz spaces was the key factor in the proof of the main result). In this paper we consider the Lorentz and Marcinkiewicz sequence spaces generated by decreasing weight sequences. In the first two sections we shall characterize the smooth and extreme points of the balls in these spaces. In the last section we shall apply these results to study the relationship between the existence and one-complemented subspaces of Lorentz sequence spaces. Let’s first agree on basic definitions and notations. Throughout the paper any vector space will be always considered over the field of real numbers R. Given a Banach space (X, ‖ · ‖), by SX and BX we denote the unit sphere and the unit ball of X, respectively. Recall that x ∈ SX is an extreme Department of Mathematics, POSTECH, San 31, Hyoja-dong, Nam-gu, Pohang-shi, Kyungbuk, Republic of Korea Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland 1 2 HAN JU LEE1,ANNA KAMIŃSKA2 AND GRZEGORZ LEWICKI3 point of the ball BX whenever x = (x1 + x2)/2 with xi ∈ SX , i = 1, 2, implies that x = x1 = x2. An element x ∈ X is called a smooth point of X if there exists a unique bounded linear functional φ ∈ SX∗ such that φ(x) = ‖x‖. Such functional φ is a called a supporting functional of x. A symbol extC will stand for the set of all extreme points of a convex subset C of X. Assume that {w(n)} is a decreasing sequence of positive numbers such that limn w(n) = 0 and ∑∞ n=1 w(n) = ∞. Let W (n) = ∑n i=1 w(i). By cardA we denote cardinality of A ⊂ N. For a real sequence x = {x(n)}, by x∗ = {x∗(n)} we denote its decreasing rearrangement. Recall that x∗(n) = inf{s > 0 : dx(s) ≤ n}, n ∈ N, where dx is a distribution of x, that is dx(s) = card{k ∈ N : |x(k)| > s}, s ≥ 0. For any x = {x(n)} the support of x is the set suppx = {n ∈ N : x(n) 6= 0}. We say that two sequences are equimeasurable whenever their distributions coincide. The Lorentz sequence space d(w, 1) is a collection of all real sequences x = {x(n)} such that

Extreme and Smooth Points in Lorentz and Marcinkiewicz Spaces with Applications to Contractive Projections