Prove: Integral Inequality for Convex Function

• MHB
• anemone
In summary, the Integral Inequality for Convex Functions is a theorem that states the integral of a convex function over a closed interval is greater than or equal to the product of the interval's length and the average value of the function. It is typically proven using the Mean Value Theorem and has applications in optimization, economics, engineering, and physics. It can also be extended to multiple dimensions and has various versions for different types of convex functions.
anemone
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MHB
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Let $f:[1,\,13]\rightarrow R$ be a convex and integrable function. Prove that $\displaystyle \int_1^3 f(x)dx+\int_{11}^{13} f(x)dx\ge \int_5^9 f(x)dx$,

If $a<b<c$, we then have $f(a-b+c)+f(b)\le f(a)+f(c)$.

Let $c=a+10$ and $b=a+4$. Then we have

$f(a+6)+f(a+4)\le f(a)+f(a+10)$

If we integrate both sides for $a\in [1,\,3]$, and notice that

$\displaystyle \int_1^3 f(a+6)da=\int_{7}^{9} f(x)dx,\\ \displaystyle\int_1^3 f(a+4)da=\int_{5}^{7} f(x)dx$
and
$\displaystyle \int_1^3 f(a+10)da=\int_{11}^{13} f(x)dx$

the result then follows.

1. What is the definition of a convex function?

A convex function is a function where the line segment connecting any two points on the graph of the function lies above or on the graph itself. In other words, the function is always "curving upwards" and never "curving downwards."

2. What is the integral inequality for convex functions?

The integral inequality for convex functions states that for any convex function f(x) on an interval [a,b], the following inequality holds: ∫ab f(x) dx ≥ (b-a) * f((a+b)/2). This means that the integral of a convex function over an interval is always greater than or equal to the rectangle formed by the interval's width and the function's average value over that interval.

3. Can you provide an example of a convex function and its integral inequality?

One example of a convex function is f(x) = x2 on the interval [0,2]. The integral inequality for this function would be: ∫02 x2 dx ≥ (2-0) * (f(0) + f(2))/2 = 2 * (0 + 4)/2 = 4. This means that the integral of f(x) = x2 over [0,2] is greater than or equal to 4.

4. How is the integral inequality for convex functions useful in mathematics?

The integral inequality for convex functions is useful in many areas of mathematics, such as optimization, analysis, and geometry. It allows us to make conclusions about the behavior of a convex function over an interval without needing to know the exact values of the function at every point in that interval. This can simplify and speed up calculations in various mathematical problems.

5. Can the integral inequality for convex functions be extended to higher dimensions?

Yes, the integral inequality for convex functions can be extended to higher dimensions. In higher dimensions, the integral inequality becomes a volume inequality, where the integral of a convex function over a region is always greater than or equal to the volume of the region multiplied by the average value of the function over that region. This is known as the Brunn-Minkowski inequality and has many applications in geometry and optimization.

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