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In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure of m-dimensional area-minimizing surfaces in R. Calibrations correspond to faces of the Grassmannian G( m, R) of oriented m-planes in R, viewed as a compact submanifold of the exterior algebra Λ R. Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands Φ more general than area. One result implies that for m-dimensional surfaces in R, with 2 ⩽ m ⩽ n − 2, for any integrand Φ, there are Φ-minimizing surfaces with interior singularities. [ABSTRACT FROM AUTHOR]