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I was reading this lecture on convex functions and I came across this

$f\colon \Bbb R^n\to \Bbb R$ is convex if and only if the function $g\colon \Bbb R\to \Bbb R$, $g(t) = f(x+tv)$, $\operatorname{dom}g=\{t\mid x+tv \mbox{ belongs to domain of }f\}$ is convex in $t$ for any $x$ belongs to the domain of $f$, $v$ belongs to $\Bbb R^n$.

So, I am a bit confused how this function of line is chosen. I mean $t$ and $v$ both are arbitrary as mentioned by the lecturer. But I want to visualize how this line looks like. I mean does it lie in the function itself.

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    The idea is that if I have to check convexity of a function, I just have take a plane and slice the function in every which way possible and see if that slice looks convex or not. Since that slice is just a curve, it is very easy to check for convexity. The g(t) parameterization captures this notion of the slice of the function.2012-06-17
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    can you elaborate what you mean by slicing a function? I mean how can I get a line when I slice a function by a plane?2012-06-17
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    When I say slice, I mean look at the intersection of that plane with the function.2012-06-17
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    I didn't get it. When I get the intersection, how come that is a line then? I could be anything for eg it could be a circle?2012-06-17
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    I mean the lecturer said that it is restriction of a convex function to a line. Does it mean that we will select an arbitrary line and view the function and see if that satisfies the convexity. But I am not sure how g(t) is a line. It is just a function of t. But it could be non linear as well. So how come it is a line?2012-06-17
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    >> Does it mean that we will select an arbitrary line ## Yes. >> But I am not sure how g(t) is a line. It is just a function of t. But it could be non linear as well. So how come it is a line? ## g(t) isn't a line, x + tv is the line.2012-06-17
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    So x+tv is a line just like Ax+b = 0 equation. But still. I didn't get after using g(t) are we checking the convexity of a totally different function g(t). I don't get how g(t) and f are related? Can you give a good example?2012-06-17
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    Ok, consider a well curve in 3D. Now, its domain is a 2D plane. Now, draw a line on this 2D domain and look at the function values only along this line. That will be a curve (It probably will look like a vertical parabola). Now, the theorem is if you can show that for all such lines in the domain, if the resulting curve of f is convex, then the entire f is convex and vice versa.2012-06-17
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    Also how come x+tv is a line. v and x both are vectors of dimension n. So how come x+tv is a line?2012-06-17
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    Again, consider a usual 2D plane. Take two points x = [1 1] and v = [2 2]. Now, consider x + tv and vary the value of t, what do you see?2012-06-17

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