= Solution
Assume $K$ is nonempty; without that necessary hypothesis the assertion is false. Put $d=\inf_{z\in K}\|x-z\|_p$ and choose $y_n\in K$ with $\|x-y_n\|_p\to d$. Convexity gives $(y_n+y_m)/2\in K$, so its distance to $x$ is at least $d$. Apply the supplied uniform-convexity inequality to $x-y_n$ and $x-y_m$:
$$
\|y_n-y_m\|_p^p\leq2^{p-1}(\|x-y_n\|_p^p+\|x-y_m\|_p^p)-2^pd^p\longrightarrow0.
$$
Thus the minimizing sequence is Cauchy. Completeness of the <Banach space> $\ell^p$ gives $y_n\to y$; closedness gives $y\in K$, and continuity of the norm gives $\|x-y\|_p=d$. \b[The minimum distance is attained.] This is the <nearest point in a uniformly convex Banach space> property. The same inequality applied to two minimizers shows uniqueness, though existence alone was requested.
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