Let \(Q\), \(K \subset \mathbb{R}^N\) be compact convex sets with \(K \subset Q\), and let \(H_K\) be the support function of \(K\) defined on \(\mathbb{R}^N\) by \(H_K (y) = \sup_{x \in K} \langle x,y \rangle\). An algebraic variety \(V \subset \mathbb{C}^N\) is said to satisfy the Phragmén-Lindelöf condition \(PL (K,Q)\) if for each \(k \geq 1\), there exist \(l \geq 1\) and \(C > 0\), such that for all plurisubharmonic functions \(u\) on \(V\), the conditions (1) and (2) imply (3), where (1) \(u(z) \leq H_K ({\mathcal F} (z)) + O (\log (1 + |z |))\), \(z \in V\), (2) \(u(z) \leq H_Q ({\mathcal F} (z)) + k \log (1 + |z |)\), \(z \in V\), and (3) \(u(z) \leq H_K ({\mathcal F} (z)) + l \log (1 + |z |) + C\), \(z \in V\). The algebraic variety \(V\) satisfies \(APL (K,Q)\) if the above implications hold for all plurisubharmonic functions \(u = \log |f |\), where \(f\) is a holomorphic function on \(V\). The main result of the paper (Theorem 10) is that if \(V\) is a pure \(k\)-dimensional variety in \(\mathbb{C}^N\), and \(K \subset Q \subset \mathbb{R}^N\) are compact convex sets with nonempty interior, then \(PL (K,Q)\) is equivalent to \(APL (K,Q)\).