Padoa's inequality

The inequality of Padoa ( English Padoa's inequality ) is a fundamental inequality of triangular geometry . It goes on the Italian mathematician Alessandro Padoa and became from that in 1925 published . The inequality relates two products formed from the lengths of the sides of a triangle and is equivalent to Euler's triangle inequality .

Representation of the inequality

Padoa's inequality states the following:

If any triangle is given in the Euclidean plane and its sides have the same lengths , then the inequality always applies ${\displaystyle ABC}$ ${\displaystyle a,b,c}$

(P)  . ${\displaystyle abc\geq \left(a+b-c\right)\left(b+c-a\right)\left(c+a-b\right)}$

Notes on the evidence

Following Alsina and Nelsen, one can derive the inequality of Padoa using the so-called Ravi substitution with the help of the inequality of the arithmetic and geometric mean .

The Ravi substitution begins with the fact that each of the three sides by the inscribed circle common tangent point into two partial stretch is divided, wherein at each vertex of the two there inzidierenden are sections of equal length. Taking these lengths, so one has positive numbers with ${\displaystyle ABC}$ ${\displaystyle x,y,z}$

${\displaystyle a=x+y}$

${\displaystyle b=y+z}$

${\displaystyle c=z+x}$  .

This can be used to reduce Padoa's inequality in the form

(P ^' ) ${\displaystyle \left(x+y\right)\left(y+z\right)\left(z+x\right)\geq 8xyz}$

write.

Now, however, according to the inequality of the arithmetic and geometric mean

${\displaystyle x+y\geq 2{\sqrt {xy}}}$

${\displaystyle y+z\geq 2{\sqrt {yz}}}$

${\displaystyle z+x\geq 2{\sqrt {zx}}}$

and by multiplying the respective left and right sides and taking into account the monotony laws for inequalities one immediately obtains (P ^' ) and thus (P) .

Equivalence with Euler's inequality

The fact that the Padoe and Euler's inequalities are equivalent can be reduced to three basic equations. Namely, by designating the circumcircle or incircle radius in the triangle with or as well as with its area and thereby , one obtains through elementary geometric considerations ${\displaystyle ABC}$ ${\displaystyle R}$ ${\displaystyle r}$ ${\displaystyle F}$ ${\displaystyle s=x+y+z}$

(G ₁ ) ${\displaystyle 4FR=abc}$

(G ₂ ) ${\displaystyle r^{2}s=xyz}$

(G ₃ ) ${\displaystyle F=rs}$

and from this the equivalence of the two inequalities.

Related inequalities

With the same names as above one also has:

(V ₁ ) ${\displaystyle {\sqrt {a}}+{\sqrt {b}}+{\sqrt {c}}\geq {\sqrt {a+b-c}}+{\sqrt {b+c-a}}+{\sqrt {c+a-b}}}$

(V ₂ ) ${\displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}\leq {\frac {1}{a+b-c}}+{\frac {1}{b+c-a}}+{\frac {1}{c+a-b}}}$

literature

• Claudi Alsina , Roger B. Nelsen : When Less is More: Visualizing Basic Inequalities (=  The Dolciani Mathematical Expositions . Volume 36 ). The Mathematical Association of America , Washington, DC 2009, ISBN 978-0-88385-342-9 ( MR2498836 ).
• Albert W. Marshall , Ingram Olkin : Inequalities: Theory and Majorization and Its Applications . Academic Press , New York, London, Toronto, Sydney, San Francisco 1979, ISBN 0-12-473750-1 ( MR0552278 ).
• A. Padoa: Una questione di minimo . In: Periodico di Matematiche . tape 4 , 1925, pp. 80-85 .

Individual evidence

1. ^ Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. 2009, pp. 14, 58, 176
2. ^ Albert W. Marshall, Ingram Olkin: Inequalities: Theory and Majorization and Its Applications. 1979, p. 202
3. ↑ Alsina / Nelsen, op.cit., P. 13
4. ↑ Apparently with half the size of is identical. ${\displaystyle s=x+y+z}$ ${\displaystyle ABC}$
5. ↑ (G ₂ ) is equivalent to Heron's formula .
6. ↑ Alsina / Nelsen, op.cit., P. 58
7. ↑ Alsina / Nelsen, op.cit., P. 14
8. ↑ Marshall / Olkin, op.cit., P. 202
9. ↑ Here an inequality given by Marshall / Olkin was simplified by algebraic transformations .
10. Jump up ↑ Ingram Olkin (July 23, 1924 - April 28, 2016) was an eminent American statistician. See article Ingram Olkin in the English language Wikipedia!