Two inequalities: a geometric and a combinatorial

Abstract

We present two interesting inequalities: one geometric and one combinatorial. The geometric one involves symmetric functions of side lengths of a triangle. It simultaneously improves Euler's inequality and isoperimetric inequality for triangles and has non-Euclidean versions. As a consequence, in combinatorics we apply it to degenerate (Fibonacci) triangles. We discuss similar inequalities for simplices in higher dimensions. The combinatorial inequality deals with the following question. What is more probable among maps: an injection or a surjection? For maps between finite sets, the answer is surjection. We present several proofs and provide a brief discussion on open problems for continuous maps for metric and other spaces.Keywords: triangle inequality, tetrahedron and volume inequality, Euler's inequality in 2D and 3D, combinatorial inequality, injective proof MSC: 51M04, 51M09, 51M16, 05A20, 60C05

Refinements of Euler's inequalities in plane, space and n-space

“…This is also explained in [1] and [5]. The left inequality in (5) improves the inequality S ≤ 3(abc) 2/3 4 , which, in turn, improves the standard isoperimetric inequality for triangles, as proved by Theorem 4.1. in [11]. So, we have the following chain of bounds for the area S of an (a, b, c)-triangle:…”

“…and this inequality is obviously equivalent to the left inequality in (5). The right inequality in (5) follows from the AM-GM inequality abc ≤ ( a+b+c 3 ) 3 and the inequality abc abc+a 3 +b 3 +c 3 ≤ 1/4, and this is equivalent to the AM-GM inequality abc ≤ a 3 +b 3 +c 3…”

“…The inequality (18) was proved via computer aided argument and by quantum information theory in 2015 and 2018 (see lit. in [5]). So, the interest for finding a proof of (18) even among computer scientists shows its importance.…”

“…Proof of (2) is given in [1]; one algebraic and one geometric proof; see also [5]. We provide here a short algebraic proof.…”

“…Using this Lemma, the inequality (2) or (4) can easily be converted into corresponding spherical or hyperbolic analogues. This is elaborated in more details in [1] and [5]. For instance, in hyperbolic geometry, the inequality (4) holds with symmetric functions in variables sinh(a/2) etc.…”