Related to the convexity of the following function?

Question

I have a function $f(x)$ which is convex with respect to $x$. Suppose if I introduce a new variable $y=x+c$ where $c$ is some positive constant then will $f(y)=f(x+c)$ be also convex? Thanks in advance.

Answer 1

Graphically, you are just translating the graph horizontally, so yes, it is convex.

Suppose $f(x)$ is convex, now let $g(x)=f(x+c)$,

\begin{align} g( \lambda x_1 + (1-\lambda) x_2) &= f( \lambda x_1 + (1-\lambda) x_2+c) \\ &=f( \lambda (x_1+c) + (1-\lambda) (x_2+c)) \\ &\le \lambda f(x_1+c)+(1-\lambda)f(x_2+c) \\ &=\lambda g(x_1) + (1-\lambda) g(x_2) \end{align}

Answer 2

More generally, if $A$ is linear, then the function $g(x) = f(Ax)$ is convex.

$g(t x_1+(1-t)x_2) = f(t Ax_1 + (1-t) A x_2) \le t f(Ax_1) + (1-t) f(A x_2) = t g(x_1)+ (1-t)g(x_2) $