arXiv:2412.08484v2 Announce Type: replace-cross Abstract: Modern geometric generation methods rely heavily on deep learning methods that, while powerful, often lack interpretability and require extensive training data. This work introduces MeshCone, a convex optimization framework for mesh enhancement from partially deformed meshes that requires no training data. We formulate the problem as a second-order cone program where vertex positions are optimized to align with target geometry while enforcing smoothness through convex edge-length regularization. Our convex relaxation enables deterministic, interpretable solutions with proven convergence properties via the Splitting Conic Solver (SCS). We demonstrate robust performance across 56 diverse object categories from ShapeNet and ThreeDScans, achieving superior refinement quality compared to classical baselines while maintaining sub-second inference times. This work establishes a principled baseline demonstrating what convex optimization alone can achieve, providing mathematical guarantees and interpretability that complement data-driven approaches.