= Solution
Write the nonzero highest weight of $V$ as
$$
\lambda=\sum_{j=1}^{\ell}m_j\omega_j,
\qquad m_j\in\mathbb Z_{\geq0}.
$$
Choose $i$ with $m_i>0$. Then $\lambda-\omega_i$ is dominant. Every positive coroot is a nonnegative combination of simple coroots, so every numerator in the Weyl dimension formula for $\lambda$ is at least the corresponding numerator for $\omega_i$. Thus
$$
\dim V(\lambda)\geq\dim V(\omega_i).
$$
Minimality of $\dim V$ forces equality. If $\lambda\ne\omega_i$, some simple-coroot factor is strictly larger, making the product strict. Hence $\lambda=\omega_i$ and
$$
\boxed{V\cong V(\omega_i)},
$$
so $V$ is a <fundamental representation>.
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