  In this paper, we investigate a lower bound (say $s_{HK}(p,d)$) on
Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull
dimension $d$ with characteristic $p>0$. Especially, we focus three-dimensional
local rings. In fact, as a main result, we will prove that $s_{HK}(p,3) = 4/3$
and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity
4/3 is isomorphic to the non-degnerate quadric hyperplanes
$k[[X,Y,Z,W]]/(X^2+Y^2+Z^2+W^2)$ under mild conditions. Furthermore, we pose a
generalization of the main theorem to the case of $\dim A \ge 4$ as a
conjecture, and show that it is also true in case of $\dim A = 4$ using the
similar method as in the proof of the main theorem.
