Is the volume of a cube the greatest among rectangular-faced shapes of the same perimeter?

Is elementary solutions permitted?

$$ \frac{a+b+c }{3}\geq \sqrt[3]{abc} $$

Equality i.e. maximum volume for a given sum of side lengths is when all sides are equal

If you mean by "perimeter" the sum of the edges, then yes, the cube is the maximal rectangular parallelepiped among those with the same "perimeter".

Let the edges have lengths $(a,b,c)$.

Then the volume is $V=abc$ and the "perimeter" is $p=4(a+b+c).$

We can maximize volume while constraining the sum of the edges using Lagrange multipliers:

$$\begin{aligned} L &= abc-\lambda \left(a+b +c-\frac{p}{4}\right)\\ 0&=\frac{\partial L}{\partial a} = bc - \lambda\\ 0&=\frac{\partial L}{\partial b} = ac - \lambda\\ 0&=\frac{\partial L}{\partial c} = ab - \lambda\\ \end{aligned}$$ so that $$bc=ac=ab$$ and $$a=b=c.$$