Summary: We propose a new nonlinear positivity-preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex-centered one where the edge-centered, face-centered, and cell-centered unknowns are treated as auxiliary ones that can be computed by simple second-order and positivity-preserving interpolation algorithms. Different from most existing positivity-preserving schemes, the presented scheme is based on a special nonlinear two-point flux approximation that has a fixed stencil and does not require the convex decomposition of the co-normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star-shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so-called numerical heat-barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second-order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.