Generalizing the mode and mean like the quantile

Question

The mode, median, and means of a series of number ($x_1,x_2,...,x_N$) can be roughly thought of as the points that minimize the $p$-norm of the sequence for $p\in \{0,1,2\}$. The median is $c=\min_c\sum_i^N |x_i-c|^1$ (ignoring uniqueness of $c$). One can generalize this to get quantiles $\tau$ other than the median ($\tau=0.5$), by making the discrepancy function asymmetric. By scaling up losses on one side of $c$ compared to the other side we get $c_\tau=\min_c (1-\tau)\sum_{x_i\leq c} |x_i-c|^1+\tau \sum_{x_i>c} |x_i-c|^1$. We can make this same generalization to the mode and mean, but do these quantities have nice interpretations or usages?