Let \(G\) be a real semisimple Lie group, \(K\subset G\) a subgroup such that \(X=G/K\) is a Riemannian symmetric space of non-compact type. Let \(\mathfrak{g}=\mathfrak{k}+\mathfrak{p}\) be a Cartan decomposition of the Lie algebra \(\mathfrak{g}\) of \(G\) and \(\mathfrak{a}\subset\mathfrak{p}\) a maximal abelian subspace. Kostants’s linear convexity theorem [B. Kostant, Ann. Sci. Éc. Norm. Supér., IV. Sér. 6, 413–455 (1973; Zbl 0293.22019)] states that for any \(Y\in\mathfrak{a}\) the image of \(Ad(K)Y\) under the orthogonal projection \(\mathfrak{p}\to\mathfrak{a}\) equals the convex hull \(\text{conv}({\mathcal W}\,Y)\) of the orbit of \(Y\) under the Weyl group \({\mathcal W}\). This paper gives complex versions of convexity results related to this. Consider the complexification \(X_{\mathbb{C}}=G_{\mathbb{C}}/K_{\mathbb{C}}\) of \(X\), and a base point \(x_o\in X_{\mathbb{C}}\). Let \(\Sigma\) be the root system of \(\mathfrak{g}\) with respect to \(\mathfrak{a}\) and \(\Omega\subset\mathfrak{a}\) the convex set defined by the inequalities \(\left| \alpha(Y)\right| <\pi/2\) for all \(\alpha\in\Sigma\). The complex crown of \(X\) is the saturation of \(\exp(i\,\Omega).x_o\subset X_{\mathbb{C}}\) under the action of \(G\) on \(X_{\mathbb{C}}\). It is proved that for all \(Y\in\Omega\) the following convexity theorem holds: \[ G\exp(i\,Y).x_o\bigcap K_{\mathbb{C}}\,A_{\mathbb{C}}.x_o \subset K_{\mathbb{C}}\,A \exp \left(\text{ conv}({\mathcal W}\,Y)\right). \] For the proof, results of the author and R. Stanton to be published in forthcoming papers are used, namely spherical functions \(\varphi_\lambda\) on \(X\) (resp. the heat kernel \(k_t\) of \(X\)) admit holomorphic extensions \(\widetilde{\varphi}_\lambda\) (resp. \(\widetilde{k}_t\)) to the complex crown and \(K_{\mathbb{C}}\)-invariant holomorphic extensions to \(K_{\mathbb{C}}\,A,\exp(i\,\Omega).x_o\). Estimates of \(\widetilde{\varphi}_\lambda(a)\) for \(a\in \exp(i\,\Omega)\) and large parameters \(\lambda\) are leading to the result. As an application to harmonic analysis, it is proved that \(\widetilde{\varphi}_\lambda\) is bounded. Moreover, the holomorphic extension \(\widetilde{k}_t\) of the heat kernel \(k_t\) of \(X\) to the complex crown is bounded.