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We study functions defined in the plane E 2 in which level curves are strictly convex, and investigate area properties of regions cut off by chords on the level curves. In this paper we give a partial answer to the question: Which function has level curves whose tangent lines cut off from a level curve segment of constant area? In the results, we give some characterization theorems regarding conic sections.

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations on space-times M = (a, ∞) × f N with prescribed scalar curvature functions.