An n dimensional even function

Question

I am now investigate some behavior of n-dimensional even functions on $\Bbb R^n$.

For 1-dimensional even functions, because $f(x) = f(-x)$ for all $x \in \Bbb R$, so we only have to investigate for $x \geq 0$.

For 2-dimensional even functions, because $f(x_1, x_2) = f(-x_1, -x_2)$ for all $x = (x_1, x_2) \in \Bbb R^2$, we only have to investigate for the region $R = \{ (x_1, x_2) \in \Bbb R^2 \; | \;x_2 \geq -x_1 \}. $

Then how can I generalize this for $n$ dimensional even functions? (I mean how can I genralize the region $R$ above)

Please help!

Answer 1

I interpret your question as asking for a minimal region which uniquely defines an even function. In that case, your region for $n=2$ contains points it shouldn't: it contains both $(-1,1)$ and $(1,-1)$, but these map to the same value for even functions. For general $n$, take the set of $x=(x_1,\ldots,x_n)$ such that the first nonzero coordinate of $x$ is positive. For any nonzero $x\in\mathbb R^n$, this region contains either $x$ or $-x$, but not both. For $n=2$ this gives you the union of a half-plane and a half-line: $\{(x_1,x_2)\in\mathbb R^2: x_1>0$ or $(x_1=0$ and $x_2>0)$ or $(x_1=0$ and $x_2=0)\}$.