The author introduces and studies new binary operations for convex sets \(K_0,K_1\) in a finite-dimensional real vector space \(V\). They refer to a fixed decomposition of \(V\) into complementary linear subspaces \(L\) and \(M\). If \(x\in V\), \(x=y+z\) with \(y\in L\) and \(z\in M\), write \(X=(y,z)\). The fibre sum of \(K_0\) and \(K_1\) relative to \((L,M)\) is defined by \[ K_0\uplus K_1:=\bigl\{ (y_0+y_1,z) \mid(y_i,z)\in K_i\text{ for }i=0,1\bigr\}. \] Thus \(\uplus\) interpolates between Minkowski sum (case \(L=V)\) and intersection (case \(M=V)\). \(K_0\uplus K_1\) is again convex. Among the results of the investigation are topological properties, an extension of the Brunn-Minkowski theorem, a representation of the support function of \(K_0\uplus K_1\), the determination of the polar body of \(K_0\uplus K_1\) if \(o\) is in the interiors of \(K_0\) and \(K_1\). The latter result is the motivation for introducing a further binary operation for convex sets. The paper concludes with the study of a related operation for convex functions.