This paper is a thorough study of the notion of Lie semialgebras and of their analytic geometry. A Lie semialgebra is a wedge (closed convex cone) W of a finite dimensional Lie algebra L which is locally closed under the local Campbell-Hausdorff multiplication * in L. Lie semialgebras arise as the tangent objects of local subsemigroups of Lie groups which are generated by local one-parameter semigroups. (For this and a general introduction into the foundations of Lie semigroups, see the article by \textit{K. H. Hofmann} and \textit{J. D. Lawson} [Proc. Conf., Oberwolfach 1981, Lect. Notes Math. 988, 128-201 (1983; Zbl 0524.22003)]. For example, the wedges which are invariant under the adjoint representation of L are of this kind, by results of \textit{G. I. Ol'shanskij} [Funct. Anal. Appl. 15, 275-285 (1981; Zbl 0503.22011), Sov. Math., Dokl. 26, 97-101 (1982; Zbl 0512.22012)]. The eventual aim of the paper is to show that the given definition of a Lie semialgebra does not depend on a specific Campbell-Hausdorff neighborhood, and that the Lie semialgebras are characterized among the wedges in L by the property that \([x,T_ x]\subseteq T_ x\) for all \(x\in W\) allowing a unique tangent hyperplane \(T_ x\). Several interesting applications of this result are also presented. One of them is a direct proof of the mentioned result of Ol'shanskii; another one is the result that in an exponential Lie algebra L (in which there is a global analytic extension of the Campbell-Hausdorff multiplication *) a Lie semialgebra \(W\subseteq L\) is globally closed under *. In {\S} 1, along with the technical preparations for the geometric ingredients of the theory of Lie semialgebras, the authors develop in detail a complete duality theory for wedges in (not necessarily finite- dimensional) locally convex topological vector spaces. This theory contains a wealth of information of independent interest; a central result of it is a Galois correspondence between the exposed faces of a wedge W and the exposed faces of the dual wedge \(W^*\). - The duality is then combined with the notion of the tangent space of W at a point x (in an infinitesimal sense), consisting of all two-sided tangent vectors of W in X, i.e. vectors v for which there are sequences \(x_ n^+,x_ n^- \) in W such that \(\pm v=n\cdot (x_ n^{\pm}-x).\) The tangent hyperplanes (i.e. hyperplanes which are tangent spaces) play a special role; they can be characterized geometrically as unique support hyperplanes through duality by relating them to exposed faces of \(W^*.\) In {\S} 2, the main theorems, which we have already mentioned, are then derived. Together with the geometric results of {\S} 1 they also imply that for a Lie semialgebra \(W\subseteq L\) and a tangent hyperplane T the subvectorspace generated by \(T\cap W\) is a Lie subalgebra in L. T itself need not be a Lie subalgebra. Only in low dimensions (dim \(L\leq 3)\) this is also true. This fact is used already in the introduction for a classification of all Lie semialgebras of dimension \(\leq 3.\) On the other hand, {\S} 3 describes a construction which yields examples to the contrary. If L is a Lie algebra with invariant compatible norm (e.g. a compact Lie algebra), then it is shown as a consequence of the results of this paper that \(W=\{(x,r);\| x\| \leq r\}\) is a Lie semialgebra in \(L\times {\mathbb{R}}\) which generates \(L\times {\mathbb{R}}\) as a vector space. If L is compact semisimple, no tangent space of W is a subalgebra. The construction shows, in particular, that in a compact Lie algebra with nontrivial center one can always find generating Lie semialgebras. In contrast, Lie semialgebras in compact semisimple Lie algebras are never generating (by a result of the authors to be published in Geometriae Dedicata). \(\{\) In the last sentence, the word ''semisimple'' has been inadvertently omitted in the original text of the paper, creating a contradictory statement. This misprint has been communicated to the reviewer by the second author, together with the following: In Theorem 1.17, statement (iv), the right hand side of the given description of \(T_ x^{\perp}\) must be replaced by its topological closure in case L is infinite dimensional. In Theorem 1.20, statement (2) should read \(\Phi \in EXP(W^*)\). In Lemma 1.24, \(0\neq \omega \in x^{\perp}W^*\). A bibliographical indication: the papers [HH1], [HH2] and [HH3] will appear in Manuscripta Math., J. Funct. Analysis and Geometriae Dedicata, resp.\(\}\).