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Let n # 1 and B # 2. A real-valued function f defined on the n-simplex #n is approximately convex with respect to #B-1 if f # B # i=1 t i x i # # B # i=1 t i f(x i ) + 1 for all x1 , . . . , xB # #n and all (t1 , . . . , t B ) # #B-1 . We determine the extremal function of this type which vanishes on the vertices of #n . We also prove a stability theorem of Hyers-Ulam type which yields as a special case the best constants in the Hyers-Ulam stability theorem for #-convex functions

Year: 2000

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