See Spring ‘13 CMU Data Mining: Lecture 21 by Ryan Tibshirani, slides 4 and 5.
With sphered $x$ and $\mu_k$, the LDA discriminant function becomes distance to the mean (plus log prior):
\[\begin{align*} \hat{\delta_1}(x) &= -\frac{1}{2} (x - \hat{\mu_1})^T \Sigma^{-1} (x - \hat{\mu_1}) + \log \hat{\pi_1}\\ &= -\frac{1}{2} (x - \hat{\mu_1})^T UD^{-1}U^T (x - \hat{\mu_1}) + \log \hat{\pi_1} \\ &= -\frac{1}{2} \lVert D^{-\frac{1}{2}} U^T (x - \hat{\mu_1}) \rVert^2 + \log \hat{\pi_1}\\ &= -\frac{1}{2} \lVert x^* - \mu_1^* \rVert^2 + \log \hat{\pi_1}\\ \end{align*}\]Hence, to maximize the discriminant function, we minimize the distance to the centroid, modulo the log prior.