Minimal Hilbert-Kunz multiplicity

In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I)-e_{HK}(I^{\mathrm{\prime }})$ for ideals $I^{\mathrm{\prime }}\supseteq I$ with $l_A(I^{\mathrm{\prime }}\mathrm{/}I)=1$? 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.