  In this paper, we ask the following question: what is the minimal value of
the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with
$l_A(I'/I) =1$? In order to answer to this question, we define the notion of
minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we
calculate this invariant for quotient singularities and for the coordinate ring
of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$,
respectively.
