It is know that to show if a function $f \colon \Re \longrightarrow \Re$, is symmetric about the origin then if f(x) = y, it is a must that f(-x) = -y.
Given a body $C$ which is represented by a set of points $P$ in the space defined by $\Re^n$ where $n \geq 1$.
So, to show that $C$ is symmetric about the origin, we need to show $$ \forall \theta \in P \text{ implies that }-\theta \in P?$$
Please advise and thanks in advance.
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$\begingroup$Yes, this is the definition of "symmetric about the origin" for sets:
A subset $S$ of a vector space is symmetric about the origin if $-x \in S$ whenever $x \in S$.
For functions, there are two kinds of symmetry about the origin:
A function $f$ (on any vector space) is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$. It is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$.
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