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It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">V</mi><mi mathvariant="normal">M</mi></mrow></semantics></math></inline-formula>s) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula>-convexity over up and down (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">D</mi></mrow></semantics></math></inline-formula>) fuzzy relation has been introduced which is known as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">D</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula>-convex fuzzy number-valued mappings (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">D</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula>-convex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">V</mi><mi mathvariant="normal">M</mi></mrow></semantics></math></inline-formula>s). We offer a thorough analysis of Hermite–Hadamard-type inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">F</mi><mi mathvariant="normal">N</mi><mi mathvariant="normal">V</mi><mi mathvariant="normal">M</mi></mrow></semantics></math></inline-formula>s that are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">U</mi><mi mathvariant="normal">D</mi></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi></mrow></semantics></math></inline-formula>-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for <i>p</i>-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples.