Motivated by the seminal work of E. M. Alfsen and E. G. Effros from the 1970s on the structure of compact convex sets and its applications to the structure of real Banach spaces, the authors consider in this article analogous concepts in complete order smooth \(\infty\)-normed spaces \(V\). If \(W\) is an order smooth subspace of \(V\), then any positive bounded linear functional on \(W\) has a norm-preserving extension to a positive linear functional on \(V\). We recall that, when \(W\) is a closed subspace, it is said to be an \(M\)-ideal if there is a linear projection \(P: V^\ast \rightarrow V^\ast\) such that \(\ker(P) = W^\bot\) and \(\|x^\ast\| = \|P(x^\ast)\|+\|x^\ast - P(x^\ast)\|\) for all \(x^\ast \in V^\ast\). If \({(W^\bot)^+}'\) denotes the complementary cone of \((W^\bot)^+\), the main result (Theorem 3.5) states that \(W\) is an \(M\)-ideal if and only if the complementary cone is convex and the positive cone in \(V^\ast\) is an \(\ell^1\)-direct sum of these two cones. Let \(Q(V)\) be the set of positive linear functionals of norm \(\le1\) (quasi-state space) and let \(S(V)\) denote the state space of unit vectors in \(Q(V)\). If \(V\) has an approximate order unit, the authors show that \(F \subset S(V)\) is a split face if and only if \(\mathrm{conv}(F \cup \{0\})\) is a split face of \(Q(V)\).