= Solution
Let the fixed positive <volume> be $V=abc$. For an interior constrained extremum, <Lagrange multipliers> applied to the energy give
$$
-\frac{2A}{a^3}+\lambda bc=0,\qquad
-\frac{2A}{b^3}+\lambda ac=0,\qquad
-\frac{2A}{c^3}+\lambda ab=0.
$$
Multiplying these equations by $a,b,c$ shows that $2A/a^2=2A/b^2=2A/c^2=\lambda V$. Since the lengths are positive, $a=b=c=V^{1/3}$.
The <arithmetic-geometric mean inequality> proves this is the <global minimum>:
$$
\frac1{a^2}+\frac1{b^2}+\frac1{c^2}\geq
3\left(\frac1{a^2b^2c^2}\right)^{1/3}=3V^{-2/3},
$$
with equality exactly for equal sides. Thus the optimal shape is a \b[cube], and $\boxed{E_{\min}=3AV^{-2/3}}$.
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