This paper introduces a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures, on arbitrary triangle meshes. The authors derive first and second order differential properties using averaging Voronoi cells and a mixed Finite-Element/Finite-Volume method, comparing them to existing formulations. These operators are closely related to the continuous case, ensuring appropriate extension from continuous to discrete settings, and respect most intrinsic properties of continuous differential operators. The estimates are shown to be optimal in accuracy under mild smoothness conditions, and their numerical quality is demonstrated. Applications include mesh smoothing, enhancement, and quality checking, as well as denoising in higher dimensions, such as for tensor images. The paper discusses the challenges of estimating geometric attributes on discrete surfaces, highlighting the lack of consensus on the most appropriate methods. Previous approaches often fail to preserve important differential geometry properties on C^0 surfaces. The authors propose a new approach based on spatial averaging, which extends continuous definitions to discrete meshes. They derive discrete operators for normal vectors, mean curvature, Gaussian curvature, principal curvatures, and principal directions for piecewise linear surfaces. These operators are consistent, accurate, robust, and simple to compute. The mean curvature normal operator is derived using a systematic approach combining finite elements and finite volumes. The integral of the mean curvature normal over a surface patch is computed using the Laplace-Beltrami operator, which is a generalization of the Laplacian from flat spaces to manifolds. The result is expressed as a simple formula, valid for any triangulation. The Gaussian curvature operator is derived using the Gauss-Bonnet theorem, which relates the integral of Gaussian curvature to the total curvature of a surface. The paper also discusses the extension of these operators to higher dimensional embedding spaces, allowing their use on vector fields, tensor images, and volume data. The operators are shown to be effective for denoising and enhancing these data types, preserving important features such as edges and discontinuities. The results demonstrate the accuracy and robustness of the proposed operators in various applications, including mesh smoothing, enhancement, and denoising.This paper introduces a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures, on arbitrary triangle meshes. The authors derive first and second order differential properties using averaging Voronoi cells and a mixed Finite-Element/Finite-Volume method, comparing them to existing formulations. These operators are closely related to the continuous case, ensuring appropriate extension from continuous to discrete settings, and respect most intrinsic properties of continuous differential operators. The estimates are shown to be optimal in accuracy under mild smoothness conditions, and their numerical quality is demonstrated. Applications include mesh smoothing, enhancement, and quality checking, as well as denoising in higher dimensions, such as for tensor images. The paper discusses the challenges of estimating geometric attributes on discrete surfaces, highlighting the lack of consensus on the most appropriate methods. Previous approaches often fail to preserve important differential geometry properties on C^0 surfaces. The authors propose a new approach based on spatial averaging, which extends continuous definitions to discrete meshes. They derive discrete operators for normal vectors, mean curvature, Gaussian curvature, principal curvatures, and principal directions for piecewise linear surfaces. These operators are consistent, accurate, robust, and simple to compute. The mean curvature normal operator is derived using a systematic approach combining finite elements and finite volumes. The integral of the mean curvature normal over a surface patch is computed using the Laplace-Beltrami operator, which is a generalization of the Laplacian from flat spaces to manifolds. The result is expressed as a simple formula, valid for any triangulation. The Gaussian curvature operator is derived using the Gauss-Bonnet theorem, which relates the integral of Gaussian curvature to the total curvature of a surface. The paper also discusses the extension of these operators to higher dimensional embedding spaces, allowing their use on vector fields, tensor images, and volume data. The operators are shown to be effective for denoising and enhancing these data types, preserving important features such as edges and discontinuities. The results demonstrate the accuracy and robustness of the proposed operators in various applications, including mesh smoothing, enhancement, and denoising.