<abstract> <p>The objective of the authors is to introduce the new class of convex fuzzy-interval-valued functions (convex-FIVFs), which is known as $ p $-convex fuzzy-interval-valued functions ($ p $-convex-FIVFs). Some of the basic properties of the proposed fuzzy-interval-valued functions are also studied. With the help of $ p $-convex FIVFs, we have presented some Hermite-Hadamard type inequalities ($ H-H $ type inequalities), where the integrands are FIVFs. Moreover, we have also proved the Hermite-Hadamard-Fejér type inequality ($ H-H $ Fejér type inequality) for $ p $-convex-FIVFs. To prove the validity of main results, we have provided some useful examples. We have also established some discrete form of Jense's type inequality and Schur's type inequality for $ p $-convex-FIVFs. The outcomes of this paper are generalizations and refinements of different results which are proved in literature. These results and different approaches may open new direction for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>