The Orlicz-Brunn-Minkowski theory was introduced by Lutwak, Yang and Zhang, being an extension of the classical Brunn-Minkowski theory. It represents a generalization of the \(L_p\)-Brunn-Minkowski theory. For a convex, strictly increasing \(\phi:[0,\infty]\longrightarrow [0,\infty)\), with \(\phi(0)=0\) and \(K,L\) convex and compact sets containing the origin in their interiors, the authors define the Orlicz sum \(K+_{\phi} L\) as the convex body whose support function is given by \[ h_{K+_\phi L}(u)=\inf\left\{\tau>0:\phi\left(\frac{h_K(u)}{\tau}\right)+\phi\left(\frac{h_L(u)}{\tau}\right)\leq 1\right\}. \] The above defined Orlicz sum of two convex bodies is an important special case of a more general Orlicz sum, defined implicitly by means of its support function [\textit{R. J. Gardner} et al., J. Differ. Geom. 97, No. 3, 427--476 (2014; Zbl 1303.52002)], as \[ 1=\varphi\left(\frac{h_K(u)}{h_{K+_\varphi L}(u)},\frac{h_L(u)}{h_{K+_\varphi L}(u)}\right), \] a suitable \(\varphi\) (see [\textit{E. Lutwak} et al., Duke Math. J. 112, No. 1, 59--81 (2002; Zbl 1021.52008); J. Differ. Geom. 62, No. 1, 17--38 (2002; Zbl 1073.46027); J. Differ. Geom. 68, No. 1, 159--184 (2004; Zbl 1119.52006)], together with Theorem 5.5 in [loc. cit.]), and, ultimately, a special case of the \(M\)-sum of two convex bodies for a (suitable) planar set \(M\) (see [loc. cit.], Theorem 5.3). In this paper, the authors prove the following \textit{Orlicz Brunn-Minkowski inequality}: \[ \phi\left(\frac{V(K)^{1/n}}{V(K+_{\phi}L)^{1/n}}\right)+\phi\left(\frac{V(L)^{1/n}}{V(K+_{\phi}L)^{1/n}}\right)\leq 1 \] for \(K,L\) convex bodies containing the origin in their interior, and \(V\) the volume (see also Corollary 7.5 in [loc. cit.]). The authors prove the above inequality using, among others, properties of \textit{Orlicz combinations}. For \(\alpha\geq0\), \(\beta>0\) and \(\phi:[0,\infty)\to [0,\infty)\) convex, strictly increasing and satisfying \(\phi(0)=0\), and convex bodies \(K,L\) containing the origin in their interiors, if the following function is a support function, the convex body having it as support function is called the Orlicz combination of \(K\) and \(L\) with coefficients \(\alpha,\beta\) \[ h_{K+_\phi L}(u)=\inf\left\{\,\tau>0:\alpha\phi\left(\frac{h_K(u)}{\tau}\right)+\beta\phi\left(\frac{h_L(u)}{\tau}\right)\leq 1\right\}. \] If \(\phi(1)=1\), the authors notice that the above function is always a support function. Thus, the above Orlicz Brunn-Minkowski inequality is also a particular case of the more general inequality proven in Corollary 7.5 of [loc. cit.]. The proofs, though, are different. The authors prove, using the above Orlicz Brunn-Minkowski inequality, an extension of the \(L_p\) Minkowski mixed volume to the Orlicz mixed volume inequality, also contained in [loc. cit.]. For that, they define, via a variational quotient, the Orlicz mixed volume of two convex bodies providing an integral representation for this. This agrees also with some more general results contained in the already mentioned work by R. Gardner, D. Hug and W. Weil.