Spaces with large projection constants

For every prime numberk, we give an explicit construction of a complexk-dimensional spaceX _k with projection constantγ(X _k ) = √k − 1/√k + 1/k. Moreover, there are realk-dimensional spacesX _k withγ(x _K ) ≧ √k − 1 for a subsequence of integersk. Hence in both casesγ(X _k )/√k → 1 which is the maximal possible value sinceγ(X _k ) ≦ √k is generally true.