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Let $n\in \mathbb{N}$ and $\mathcal{E}_n$ the set of functions $f$ of the form $$f(x)=b_0+b_1x+\sum_{i=1}^n a_i\left\lvert x-r_i\right\rvert$$ for $b_0,b_1\in\mathbb{R},a_1,\dots,a_n>0$ and $r_1,\dots,r_n\in\mathbb{R}$.

Show that if $f$ is a convex continuous piecewise linear function, then there exists $n\in\mathbb{N}$ such that $f\in\mathcal{E}_n$.

This result does not seem clear to me, and I do not know how to prove it.

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Here is my proof for this.

Let $k_1<\dots k_n\in\mathbb{R}$ such that $f$ is linear on $\left]k_i,k_{i+1}\right[$ for all $1\leqslant i\leqslant n-1$ and $f$ is linear on $\left]-\infty,k_1\right[$ and on $\left]k_n,+\infty\right[$.

Let $c_i$ be the slope of $f$ on $\left[ k_i,k_{i+1}\right[$ for all $0\leqslant i\leqslant n$ with the convention $k_0=-\infty$ and $k_{n+1}=+\infty$.

As $f$ is convex, we have $c_0\leqslant c_1\leqslant\dots\leqslant c_{n+1}$. Let $$a_i=\frac{1}{2}\left(c_i-c_{i-1}\right)$$ for $1\leqslant i\leqslant n$. Indeed, in the formula $x\mapsto b_0+b_1+\sum_i\left\lvert x-k_i\right\rvert$, the jump of the derivative in $x=k_i$ is $2a_i$.

Let $b_1$ such that $$b_i+\sum_{i=1}^na_i=c_n$$ that is $b_1=\frac{1}{2}\left(c_n+c_o\right)$.

As such, the two applications $f$ and $x\mapsto b_1x+\sum_{i=1}^na_i\left\lvert x-k_i\right\rvert$ are two linear functions on every intervals $\left]k_{i-1},k_i\right[$ for $1\leqslant i\leqslant n+1$, continuous and with the same slope on every interval by construction.

That is the two applications differ by a constant, named $b_0$. Thus, $f\in \mathcal{E}_n$.

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