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Geometric view of Hölder's inequality
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[QUICK DESCRIPTION] To understand the standard Hölder inequality better it is helpful to look at a more general version and use a geometric picture of convex bodies. [PREREQUISITES] Undergraduate analysis. [EXAMPLE] Say, we want to bound the integral [math] \int_{0}^1 f(x)^5g(x)^2 d\,x. [/math] Our current knowledge allows us to estimate the integrals [math] \int_{0}^1 |f(x)|^4|g(x)|^2 dx, \int_{0}^1 |f(x)|^8 dx \quad and \quad \int_{0}^1 |g(x)|^8 dx. [/math] How can we see quickly, whether this information is sufficient or not? Secondly, how are we going to choose the coefficients in Hoelder's inequality? This type of problem is common in analytic number theory, for example. [GENERAL DISCUSSION] [[w:Hölder's inequality]] states that on any measure space $(X,\mu)$, and any measurable $f, g: X \to \C$, one has [math] \Big|\int_X f(x) g(x)\ d\mu\Big| \leq \Big(\int_X |f(x)|^p\ d\mu\Big)^{1/p} \Big(\int_X |g(x)|^q\ d\mu\Big)^{1/q}[/math] whenever we have $1/p+1/q = 1$. The key insight here is to view this as an statement about interpolation to get information at the point $(1,1)$ from information at the points $(p,0)$ and $(0,q)$. The condition $1/p+1/q=1$ ensures that all three points lie on a line. Once we have this realisation, it is easy to generalise this to the case of the example above. To obtain information at the point $(a,b)$, as in the integral [math] \Big|\int_X f(x)^a g(x)^b\ d\mu\Big|[/math] it is sufficient to have bounds at three points $(u_1,v_1),(u_2,v_2),(u_3,v_3)$ such that $(a,b)$ lies in the convex hull of the three points. The coefficients in the estimate are then given by the coefficients of the convex linear combination of $(a,b)$ inside the triangle $(u_1,v_1),(u_2,v_2),(u_3,v_3)$. The simplicity of the geometric view allows us to handle more complex integrals integrals like [math] \Big|\int_X f(x)^a g(x)^b h(x)^c\ d\mu\Big|[/math] with the same ease. By drawing the corresponding points in the plane, it is easy to judge in advance which of the integrals contributes most to the estimate later, thus providing a tool to estimate the quality of the estimate.
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