The projection constants ofthe lpn-spaces for 1 less than or equal to p les
s than or equal to 2 satisfy lambda(lpn)/root n --> c with c = 2/pi in the
real case an c = root pi/2 in the complex case. Further, there is c < 1 suc
h that the projection constant of any n-dimensional space Xn with 1-symmetr
ic basis can be estimated by lambda(Xn) less than or equal to c root n. The
proofs of the results are based on averaging techniques over permutations
and a variant of Khintchine's inequality which states that
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