Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L1, so-called regular symmetric subspaces. Using this notion, we show that for any k⩾2, there exist k-dimensional symmetric subspaces of L1 which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L1 is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases