Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Abstract

Leta1,b1,c1,A1 anda2,b2,c2,A2 be the sides and areas of two triangles. Ifa=(a1p+a2p)1/p,b=(b1p+b2p)1/p,c=(c1p+c2p)1/p, and 1≤p≤4, thena, b, c are the sides of a triangle and its area satisfiesAp/2≥A1p/2+A2p/2. If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaAf, which satisfies (4Af/√3)1/2≥f((4A/√3)1/2).