Density of a semigroup in a Banach space

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We study conditions on a set in a Banach space which are necessary or sufficient for the set of all sums , , to be dense in . We distinguish conditions under which the closure is an additive subgroup of , and conditions under which this additive subgroup is dense in . In particular, we prove that if is a closed rectifiable curve in a uniformly convex and uniformly smooth Banach space , and does not lie in a closed half-space , , and is minimal in the sense that every proper subarc of lies in an open half-space , then . We apply our results to questions of approximation in various function spaces.