Generalized altitudes and their bounds

We introduce generalized altitudes of a simplex, extending the usual vertex-to-opposite-face altitude to arbitrary pairs of opposite faces. These quantities encode the relative position of the affine spans of such faces and yield a uniform formula for the angle between them. We also derive an equivalent algebraic expression in terms of generalized cross products and Gram determinants, linking the construction to standard determinant-based tools. Finally, we prove that every generalized altitude is bounded below by a quantity controlled by the ordinary height of the simplex. Thus, classical height or thickness assumptions imply control over this broader family of geometric quantities. The results provide a compact framework for studying simplex quality and are motivated by applications to triangulation criteria for Riemannian manifolds.