diff --git a/src/sets.jl b/src/sets.jl
index aaf69260bd..62698bda21 100644
--- a/src/sets.jl
+++ b/src/sets.jl
@@ -760,6 +760,9 @@ NormCone(set::NormOneCone) = NormCone(1, set.dimension)
 NormCone(set::SecondOrderCone) = NormCone(2, set.dimension)
 NormCone(set::NormInfinityCone) = NormCone(Inf, set.dimension)
 
+dual_set(s::NormCone) = NormCone(inv(1 - inv(s.p)), s.dimension)
+dual_set_type(::Type{NormCone}) = NormCone
+
 """
     GeometricMeanCone(dimension::Int)
 
diff --git a/test/General/test_sets.jl b/test/General/test_sets.jl
index cffd2d985d..0f145a0121 100644
--- a/test/General/test_sets.jl
+++ b/test/General/test_sets.jl
@@ -229,6 +229,19 @@ function test_sets_dual_norm()
     return
 end
 
+function test_sets_dual_pnorm()
+    norminf = MOI.NormCone(Inf, 3)
+    normone = MOI.NormCone(1.0, 3)
+    normtwo = MOI.NormCone(2.0, 3)
+    normfive = MOI.NormCone(5.0, 3)
+    normfivedual = MOI.NormCone(1.25, 3)
+    _dual_set_test(normone, norminf)
+    _self_dual_set_test(normtwo)
+    @test MOI.dual_set(normfive).p ≈ normfivedual.p
+    @test MOI.dual_set(normfivedual).p ≈ normfive.p
+    return
+end
+
 function test_sets_dual_soc()
     soc2 = MOI.SecondOrderCone(2)
     soc3 = MOI.SecondOrderCone(3)