Support slopes and Banach-space vector balancing

For an alphabet V in a normed space, let B_V(w) denote the worst optimal discrepancy of signed sums over the order ideals of finite posets of width at most w. For every bounded finite-dimensional alphabet we identify its slope gamma(V) as the supremum, over words in V, of the least radius of a Minkowski average of their signed prefix polytopes. If r >= 1 is the span dimension and C = B_V(1), then 0 <= B_V(w) - w gamma(V) <= C min{K_r w^((r-1)/(r+1)), 120 r w^(2/3)}, where K_r depends only on r. For every finite nonempty A subset (0,infinity), we determine the least diameter delta(A) of a nonempty T subset R such that, for every x in T and a in A, at least one of x-a, x+a belongs to T, answering Barany's 1979 question on collinear balancing games. Each block C of a partition contributes 2 max C - g(C), where g(C) generates its additive group if discrete and equals zero otherwise. The minimum total over partitions is delta(A). For A={2,3,5,10} the minimum diameter is 19, while the worst minimum signed prefix range is 2 gamma(A)=18. We compute the mixed slope of disjoint block alphabets in unconditional Banach sums, including cases in which the formal coordinate profile does not define a vector in the outer space. Convergence of the chain-to-slope defect profile suffices to transfer the slope identity. Source repository: https://github.com/eulogica/support-slopes-vector-balancing