Laplace expansion

2013-06-10T03:41:43Z

130.123.104.22: /* Example */

In mathematics, in the field of [[functional analysis]], a '''Minkowski functional''' is a function that recovers a notion of distance on a linear space.

Let ''K'' be a symmetric convex body in a linear space ''V''. We define a function ''p'' on ''V'' as

:<math>p(x) = \inf \{ \lambda \in \mathbb{R}_{> 0} : x \in \lambda K \} </math>

if that [[infimum]] is well-defined.<ref>Thompson (1996) p.17</ref>

== Motivation ==
===Example 1===

Consider a [[normed vector space]] ''X'', with the norm ||·||. Let ''K'' be the unit sphere in ''X''. Define a function ''p : X →'' '''R''' by
:<math>p(x) = \inf \left\{r > 0: x \in r K \right\}. </math>

One can see that <math>p(x) = \|x\|</math>, i.e. ''p'' is just the norm on ''X''. The function ''p'' is a special case of a Minkowski functional.

=== Example 2===

Let ''X'' be a vector space without topology with underlying scalar field '''K'''. Take ''φ ∈ X' '', the algebraic dual of ''X'', i.e. ''φ : X →'' '''K''' is a linear functional on ''X''. Fix ''a > 0''. Let the set ''K'' be given by

:<math>K = \{ x \in X : | \phi(x) | \leq a \}. </math>

Again we define

:<math>p(x) = \inf \left\{r > 0: x \in r K \right\}. </math>

Then

:<math>p(x) = \frac{1}{a} | \phi(x) |.</math>

The function ''p''(''x'') is another instance of a Minkowski functional. It has the following properties:

#It is ''subadditive'': ''p''(''x'' + ''y'') ≤ ''p''(''x'') + ''p''(''y''),
#It is ''homogeneous'': for all ''α'' ∈ '''K''', ''p''(''α x'') = |''α''| ''p''(''x''),
#It is nonnegative.

Therefore ''p'' is a [[seminorm]] on ''X'', with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets. What is meant precisely by "nice" is discussed in the section below.

Notice that, in contrast to a stronger requirement for a norm, ''p(x) = 0'' need not imply ''x = 0''. In the above example, one can take a nonzero ''x'' from the kernel of ''φ''. Consequently, the resulting topology need not be [[Hausdorff space|Hausdorff]].

== Definition ==

The above examples suggest that, given a (complex or real) vector space ''X'' and a subset ''K'', one can define a corresponding Minkowski functional

:<math>p_K:X \rightarrow [0, \infty)</math>

by

:<math>p_K (x) = \inf \left\{r > 0: x \in r K \right\},</math>

which is often called the gauge of <math>K</math>.

It is implicitly assumed in this definition that 0 ∈ ''K'' and the set {''r'' > 0: ''x'' ∈ ''r K''} is nonempty. In order for ''pK'' to have the properties of a seminorm, additional restrictions must be imposed on ''K''. These conditions are listed below.

#The set ''K'' being [[convex set|convex]] implies the subadditivity of ''pK''.
#[[Homogeneous function|Homogeneity]], i.e. ''pK''(''α x'') = |''α''| ''pK''(''x'') for all ''α'', is ensured if ''K'' is ''balanced'', meaning ''α K'' ⊂ ''K'' for all |''α''| ≤ 1.

A set ''K'' with these properties is said to be [[absolutely convex set|absolutely convex]].

=== Convexity of ''K'' ===

A simple geometric argument that shows convexity of ''K'' implies subadditivity is as follows. Suppose for the moment that ''pK''(''x'') = ''pK''(''y'') = ''r''. Then for all ''ε'' > 0, we have ''x'', ''y'' ∈ (''r + ε'') ''K'' = '' K' ''. The assumption that ''K'' is convex means '' K' '' is also. Therefore ½ ''x'' + ½ ''y'' is in '' K' ''. By definition of the Minkowski functional ''pK'', one has

:<math>p_K\left( \frac{1}{2} x + \frac{1}{2} y\right) \le r + \epsilon = \frac{1}{2} p_K(x) + \frac{1}{2} p_K(y) + \epsilon .</math>

But the left hand side is ½ ''pK''(''x'' + ''y''), i.e. the above becomes

:<math>p_K(x + y) \le p_K(x) + p_K(y) + \epsilon, \quad \mbox{for all} \quad \epsilon > 0.</math>

This is the desired inequality. The general case ''pK''(''x'') > ''pK''(''y'') is obtained after the obvious modification.

'''Note''' Convexity of ''K'', together with the initial assumption that the set {''r'' > 0: ''x'' ∈ ''r K''} is nonempty, implies that ''K'' is [[absorbing set|''absorbent'']].

=== Balancedness of ''K'' ===

Notice that ''K'' being balanced implies that

:<math>\lambda x \in r K \quad \mbox{if and only if} \quad x \in \frac{r}{|\lambda|} K.</math>

Therefore

:<math>p_K (\lambda x) = \inf \left\{r > 0: \lambda x \in r K \right\}
= \inf \left\{r > 0: x \in \frac{r}{|\lambda|} K \right\}
= \inf \left\{ | \lambda | \frac{r}{ | \lambda | } > 0: x \in \frac{r}{|\lambda|} K \right\}
= |\lambda| p_K(x).
</math>

== See also ==

* [[Hadwiger's theorem]]
* [[Hugo Hadwiger]]
* [[Morphological image processing]]

==Notes==
{{reflist}}

==References==
* {{cite book | title=Minkowski Geometry | series=Encyclopedia of Mathematics and Its Applications | first=Anthony C. | last=Thompson | publisher=[[Cambridge University Press]] | year=1996 | isbn=0-521-40472-X }}

[[Category:Functional analysis]]
[[Category:Convex analysis]]

Row and column spaces

2011-08-09T02:18:33Z

130.123.96.22:

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Laplace expansion

Row and column spaces