= Solution
Let $f_i$ be the <multivariate normal density> of $N_p(\mu_i,\Sigma_i)$. Under zero-one loss, the <Bayes classifier> chooses the class with the larger posterior probability. Thus
$$
\boxed{
\delta_{\pi_0}(x)
=\mathbf1\{\pi_1f_1(x)\geq\pi_0f_0(x)\},
\qquad \pi_1=1-\pi_0.}
$$
Equivalently, it chooses class one when the log posterior odds
$$
\begin{aligned}
D_{\pi_0}(x)
={}&\log\frac{\pi_1}{\pi_0}
+\frac12\log\frac{\det\Sigma_0}{\det\Sigma_1}\\
&-\frac12(x-\mu_1)^T\Sigma_1^{-1}(x-\mu_1)
+\frac12(x-\mu_0)^T\Sigma_0^{-1}(x-\mu_0)
\end{aligned}
$$
is nonnegative. The decision boundary is $D_{\pi_0}(x)=0$.
If $\Sigma_0=\Sigma_1=\Sigma$, the $x^T\Sigma^{-1}x$ terms cancel, leaving
$$
D_{\pi_0}(x)
=(\mu_1-\mu_0)^T\Sigma^{-1}x+C,
$$
an affine function. This is the linear boundary of <linear discriminant analysis>. If $\Sigma_0\ne\Sigma_1$, the quadratic part is
$$
\frac12x^T(\Sigma_0^{-1}-\Sigma_1^{-1})x,
$$
which is nonzero, so the boundary is a possibly degenerate quadratic hypersurface, as in <quadratic discriminant analysis>. This is the <Gaussian Bayes classifier>.
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