Hornich-Hlawka inequality

The inequality of Hornich-Hlawka , sometimes just as inequality of Hlawka (Engl. Hlawka's inequality called), is a mathematical theorem at the interface between the sub-regions of linear algebra and functional analysis . The inequality goes back to the two Austrian mathematicians Hans Hornich and Edmund Hlawka and is a generalization of the triangle inequality that is valid in all Prähilbert spaces .

formulation

The theorem can be formulated as follows:

Given is a Prähilbert space over the field of real or complex numbers with the norm generated by the corresponding scalar product   ( ). ${\displaystyle X}$ ${\displaystyle \mathbb {K} }$   ${\displaystyle x\mapsto |x|={\sqrt {xx}}}$ ${\displaystyle x\in X}$

Then:

(I) Any three (not necessarily different) vectors always satisfy the inequality ${\displaystyle a,b,c\in X}$

${\displaystyle |a+b|+|b+c|+|c+a|\leq |a|+|b|+|c|+|a+b+c|}$. ( Hornich-Hlawka inequality )

(II) The inequality of Hornich-Hlawka results from the following identity of Hlawka with the help of the triangle inequality  :

${\displaystyle (|a|+|b|+|c|-|a+b|-|b+c|-|c+a|+|a+b+c|)(|a|+|b|+|c|+|a+b+c|)}$

${\displaystyle =}$

${\displaystyle (|a|+|b|-|a+b|)(|c|-|a+b|+|a+b+c|)}$

${\displaystyle +}$

${\displaystyle (|b|+|c|-|b+c|)(|a|-|b+c|+|a+b+c|)}$

${\displaystyle +}$

${\displaystyle (|c|+|a|-|c+a|)(|b|-|c+a|+|a+b+c|)}$

(III) The Hornich-Hlawka inequality includes the triangle inequality as a special case.

Evidence Sketches

Step 1 - Evidence sketch for (II)

The identity of Hlawka is gained through verification . To do this, the terms on both sides of the identity are first multiplied out , taking into account the distributive law . After deleting terms that appear on both sides, you can see that the proof of the identity to be shown is the proof of the identity

(H) ${\displaystyle {|a|}^{2}+{|b|}^{2}+{|c|}^{2}+{|a+b+c|}^{2}={|a+b|}^{2}+{|b+c|}^{2}+{|c+a|}^{2}}$

is equivalent. This in turn results from the following calculation:

${\displaystyle {\begin{aligned}{|a+b+c|}^{2}&=aa+ab+ac+ba+bb+bc+ca+cb+cc\\&={|a|}^{2}+{|b|}^{2}+{|c|}^{2}+(ab+{\overline {ba}})+(ac+{\overline {ca}})+(bc+{\overline {cb}})\\&={|a|}^{2}+{|b|}^{2}+{|c|}^{2}+2\mathrm {Re} (ab)+2\mathrm {Re} (ac)+2\mathrm {Re} (bc)\\&={|a+b|}^{2}+({|c+a|}^{2}-{|a|}^{2})+({|b+c|}^{2}-{|b|}^{2}-{|c|}^{2})\\\end{aligned}}}$

Step 2 - Evidence sketch for (I)

Since in the case of the inequality to be shown amounts to an equation, thus nothing is to be shown, it is sufficient to limit the proof to the case . ${\displaystyle c=0}$ ${\displaystyle c\neq 0}$

Here one takes into account that on the right-hand side of the already proven identity of Hlawka there are only non-negative summands , because the triangle inequality and thus also the square inequality apply , which is why all terms involved within the brackets appearing on the right turn out to be non-negative. The inference is that the product on the left is also nonnegative.

It is also to be invoiced that because of always ${\displaystyle c\neq 0}$

${\displaystyle |a|+|b|+|c|+|a+b+c|>0}$

applies. So must too

${\displaystyle |a|+|b|+|c|-|a+b|-|b+c|-|c+a|+|a+b+c|\geq 0}$

apply and thus (I) .

Step 3 - Evidence sketch for (III)

The proof results from twofold substitution . To do this, one makes for the setting with which ${\displaystyle x,y\in X}$ ${\displaystyle a=c=y,b=x-y}$

${\displaystyle 2|x|\leq |x+y|+|x-y|}$

results. Then one makes for the settlement , from which finally ${\displaystyle u,v\in X}$ ${\displaystyle x={\frac {u+v}{2}},y={\frac {v-u}{2}}}$

${\displaystyle |u+v|\leq |u|+|v|}$

and thus the triangle inequality follows.

Remarks

1. In his work from 1948, Hans Hornich showed that the Hornich-Hlawka inequality can be further generalized in Euclidean spaces . This generalization is then rather than inequality of Hornich (Engl. Inequality of Hornich ), respectively.
2. The above identity (H) is explicitly mentioned by Hornich. They (and consequently also the identity of Hlawka ) include the parallelogram equation as a special case . You win this by doing for the placement . ${\displaystyle x,y\in X}$ ${\displaystyle a=c=y,b=x-y}$

Web links

• PDF on more general inequalities
• Link to MathWorld

literature

• Hans Hornich: An inequality for vector lengths . In: Math. Z . tape 48 , 1942, pp. 268-274 ( MR0008417 ).
• Max Koecher : Linear Algebra and Analytical Geometry (=  Springer Textbook: Basic Knowledge of Mathematics ). 4th, supplemented and updated edition. Springer-Verlag, Berlin (inter alia) 1997, ISBN 3-540-62903-3 .
• DS Mitrinović : Analytic Inequalities . In cooperation with PM Vasić (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 165 ). Springer Verlag , Berlin ( inter alia ) 1970, ISBN 3-540-62903-3 ( MR0274686 ).

References and footnotes

1. ↑ Koecher: p. 177.
2. ↑ In abbreviated spelling is also written for. ${\displaystyle x,y\in X}$ ${\displaystyle xy=x\cdot y={\langle x,y\rangle }}$
3. ↑ If the underlying field is equal to the field of the real numbers , then is for always and with it . ${\displaystyle \mathbb {K} ={\mathbb {R} }}$ ${\displaystyle x,y\in X}$ ${\displaystyle xy\in {\mathbb {R} }}$ ${\displaystyle {\overline {xy}}=\mathrm {Re} (xy)=xy}$
4. ↑ Hornich: An inequality for vector lengths . In: Math. Z . tape 48 , p. 268 ff .
5. ↑ Mitrinović: pp. 172-173.
6. ↑ Hornich: An inequality for vector lengths . In: Math. Z . tape 48 , p. 274 .