On closed additive semigroups in $E\sp{n}$

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Let C(A) be the closed additive semigroup generated by a set A z En. A simple necessary and sufficient condition on A for C(A) to be a group is derived. An example which arose in the theory of random walks and stimulated these purely geometrical considerations is discussed at the end. For an arbitrary subset A of the n-dimensional Euclidean space El (n>1) let S(A) denote the smallest additive semigroup containing A, Athe closure of A with respect to the Euclidean topology, and C(A):= (S(A))-. Obviously, C(A) is the smallest closed additive semigroup containing A, i.e., the closed additive semigroup generated by A. DEFINITION. A c En is called omnilateral, if A $ 0 and for every hyperplane H through the origin with A t H there are points of A in each of the two open half-spaces produced by H. In other words, A is omnilateral if A $ 0 and u e En, x e A and ux> 0 imply the existence of an x* E A with ux* <0. Some immediate consequences are listed in the following lemma. LEMMA. (1) A is omnilateral if and only if S(A) is omnilateral. (2) A is omnilateral if and only if Ais omnilateral. (3) An additive group A is always omnilateral. (4) The image of an omnilateral set AC E under a linear transformation from En to Em is an omnilateral set in Em. We say a set A c En is genuinely n-dimensional if there is no hyperplane through the origin 0 containing A. A0 denotes the convex hull of A. THEOREM 1. A genuinely n-dimensional set A is omnilateral if and only fO is an inner point of Ac. PROOF. The if-part is trivial. Suppose, on the other hand, 0 is an exterior or a boundary point of Ac. There would be a hyperplane through Received by the editors August 31, 1971. AMS 1969 subject classifications. Primary 20M99, 50B99; Secondary lOE05, 52A20, 60J15.