Search results
Results from the WOW.Com Content Network
The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area ...
1.1 Proof by Hölder's inequality. 1.2 Proof by a direct convexity argument. ... The Minkowski inequality is the triangle inequality in (). In fact, it is a special ...
Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007). Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from P to the sides are replaced by the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross ...
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
Barrow's proof of this inequality was published in 1937, as his solution to a problem posed in the American Mathematical Monthly of proving the Erdős–Mordell inequality. [1] This result was named "Barrow's inequality" as early as 1961. [4] A simpler proof was later given by Louis J. Mordell. [5]
The Ruzsa sum triangle inequality is a corollary of the Plünnecke-Ruzsa inequality (which is in turn proved using the ordinary Ruzsa triangle inequality). Theorem (Ruzsa sum triangle inequality) — If A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are finite subsets of an abelian group, then
Kansas City Chiefs tight end Travis Kelce's father, Ed Kelce, said that he plans to only spend $10 on a gift for Taylor Swift's 35th birthday this year.
A slightly weaker version of this inequality was originally proven and published by Helmut Plünnecke (1970). [1] Imre Ruzsa (1989) [2] later published a simpler proof of the current, more general, version of the inequality. The inequality forms a crucial step in the proof of Freiman's theorem.