Limit analysis of discrete masonry assemblies, developed from Hyman’s theorem, has proven to be an effective and computationally efficient approach for analyzing masonry structures. The convex method builds on this by abstracting the interface between two rigid blocks to a single point—typically the centroid—where internal forces and moments are computed to equilibrate external loads applied to the blocks. These internal actions are constrained to prevent different types of failure, including torsion-shear failure, which defines the maximum combined tangential force and torsional moment an interface can resist. Several studies based on the convex method have formulated torsion-shear constraints, though all are limited to rectangular interfaces. Those formulations address both dry frictional contacts between blocks and cohesive interfaces within blocks, where failure can occur due to finite tensile and shear strength. In both cases, the torsion-shear constraint is highly sensitive to the geometry of the interface. To overcome this, researchers have proposed general approximation methods for torsion-shear constraints that apply across rectangular interfaces with varying aspect ratios. This paper, for the first time, explores the extension of the convex method—specifically its torsion-shear constraints—to arbitrary planar interfaces bounded by convex closed polylines. It investigates whether a single approximate constraint can be established to cover diverse geometries. The proposed formulation includes both zero-tension frictional interfaces and internal discontinuities with finite shear and tensile strength. Given the proven accuracy and efficiency of the convex method, this work represents a key step toward its application in complex assemblies with diverse interface geometries.