The closed, convex hull of every ai -summing basic sequence fails the FPP for affine nonexpansive mappings

In 2004 Dowling, Lennard and Turett showed that every non-weakly compact, closed, bounded, convex (c.b.c.) subset K of ( c 0 , ‖ ⋅ ‖ ∞ ) is such that there exists a ‖ ⋅ ‖ ∞ -nonexpansive mapping T on K that is fixed point free. This mapping T is generally not affine. It is an open question as to whether or not on every non-weakly compact, c.b.c. subset K of ( c 0 , ‖ ⋅ ‖ ∞ ) there exists an affine ‖ ⋅ ‖ ∞ -nonexpansive mapping S that is fixed point free. We prove that if a Banach space contains an asymptotically isometric (ai) c 0 -summing basic sequence ( x n ) n ∈ N , then the closed convex hull of ( x n ) n ∈ N , E : = co ¯ ( { x n : n ∈ N } ) , fails the fixed point property for affine nonexpansive mappings. Moreover, we show that there exists an affine contractive mapping U : E → E that is fixed point free. Furthermore, we prove that for all sequences b → = ( b n ) n ∈ N in R with 0 < m : = inf n ∈ N b n and M : = sup n ∈ N b n < ∞ , the closed, bounded, convex subset E = E b → of c 0 defined by E : = { ∑ n = 1 ∞ t n f n : 1 = t 1 ⩾ t 2 ⩾ ⋯ ⩾ t n ↓ n 0 } , where each f n : = b n e n , is such that there exists an affine contractive mapping U : E → E that is fixed point free.