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Minkowski's Inequality 
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#1
Dec2806, 01:18 AM

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Definitions and Useful Facts
If [itex]f : X \to \mathbb{C}[/itex] is a measurable function, define the essential supremum of f to be: [tex]f_{\infty} = \inf \{a \in [0,\infty ] : \mu (\{x : f(x) > a\}) = 0\}[/tex] where [itex]\mu[/itex] is a measure, and we adopt the convention [itex]\inf \emptyset = \infty[/itex]. Note that [tex]f_{\infty} \in \{a \in [0,\infty ] : \mu (\{x : f(x) > a\}) = 0\}[/tex] If f has finite essential supremum, we say f is an [itex]L^{\infty}[/itex] function. The set of [itex]L^{\infty}[/itex] functions forms a Banach space and [itex]._{\infty}[/itex] defines a norm on this space. So if f and g are [itex]L^{\infty}[/itex] functions, then so is f+g, and the following inequality holds: [tex]f+g_{\infty} \leq f_{\infty} + g_{\infty}[/tex] (Note: We will treat two functions as identical if the subset of the domain on which they differ has measure 0. All the terms defined above remain welldefined upon adopting this convention.) Problem When does equality hold in the above inequality? Attempt Define [itex]z : X \to C[/itex] where C is the complex circle by: [tex]z(x) = \frac{f(x)}{f(x)} \mbox{ if } f(x) \neq 0;\ z(x) = 1\mbox{ if } f(x) = 0[/tex] Then fz is a nonnegative realvalued function, and [tex]fz  f = gz  g = (f+g)z  f+g = 0[/tex] hence [tex]fz_{\infty}  f_{\infty} = gz_{\infty}  g_{\infty} = (f+g)z_{\infty}  f+g_{\infty} = 0[/tex] So assume w.l.o.g. that f is a nonnegative realvalued function. Let A and B denote the essential suprema of f and g respectively. Right now my rough idea is that we get the desired equality iff for all a < A, for all b < B, and for all c > 0, the following holds: [tex]\mu ( \{x : f(x) > a, g(x) > b, g(x)  g(x) < cg(x)\} ) > 0[/tex] It basically says that equality holds iff there is a sizeable region of the domain where f is close to its maximum, g is close to its maximum, and g is close to being a positive real. Is this right? Is there a nicer way to put it? 


#2
Dec2806, 03:43 AM

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Wanting g(x) to be close to being a positive real, where f is is nonnegative realvalued function, is the same as wanting g(x)/g(x) to be close to f(x)/f(x), where f is now just an arbitrary function. And this is important because g(x)/g(x) and f(x)/f(x) are close iff f(x) and g(x) point in pretty much the same direction (thinking of the numbers f(x) and g(x) as arrows/vectors in the complex plane) iff f(x) + g(x) is close to f(x) + g(x). So it might be neater to propose that equality holds iff:
[tex](\forall a < A)(\forall b < B)(\forall \epsilon > 0)(\mu (\{x : f(x) > a, g(x) > b, \overline{g(x)}  \overline{f(x)} < \epsilon \} ) > 0[/tex] where [itex]\overline{z} = z/z[/itex] for every nonzero complex number z. 


#3
Dec2906, 10:28 AM

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Up to accounting for the trivial case where one of the functions is a.e. zero, I think that's the best you can do. There's no nice condition as for other Lp norms, like one function being a scalar multiple of the other, since clearly the only part of the domain that matters for this condition is the set [itex]\{ x  \mbox{ }f_\inftyf(x)<\epsilon\}[/itex], any [itex]\epsilon>0[/itex], and so outside this range (which can usually be made arbitrarily small), the values of the function are completely irrelevant. Note that your condition [itex]\overline{g(x)}  \overline{f(x)} < \epsilon[/itex] is essentially a rewrite of [itex]f(x)+g(x)f(x)+g(x)<\epsilon'[/itex], some [itex]\epsilon'[/itex] that goes to zero as [itex]\epsilon[/itex] does (again, up to the case where on of the functions approaches zero), which shows your proposal is correct, if not all that enlightening.



#4
Dec2906, 11:02 AM

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Minkowski's Inequality
Also, I can easily account for the a.e. 0 case by changing the quantifiers to say [itex](\forall a \in (0,A))(\forall b \in (0,B))\dots[/itex] so if one of the functions is a.e. 0, then a or b will quantify over the empty set, making the thing trivially true. 


#5
Dec2906, 03:36 PM

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#6
Dec2906, 04:25 PM

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[tex]f+g_1 = f_1 + g_1[/tex] iff [tex]\int f+g = \int f + \int g[/tex] iff [tex]\int f+g = \int f + g[/tex] iff [tex]f+g = f + g a.e.[/tex] iff there exist nonnegative realvalued functions r and q such that they are never both zero for the same x and such that fr = gq a.e. (EDITED) 


#7
Dec2906, 06:45 PM

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Sorry, I had something backwards. It seems weird, but I guess that's right.



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