A Note on n-Subhomogeneity of Periodic Extension of Convex Functions

We prove that the T-periodic extension of a convex function

, is n-subhomogeneous if and only if

A = \lim_{x\to 0^{+}} f_{1}(x)\leq nf_{1}(k \frac{T}{n}) \quad \text{and} \quad B = \lim_{x\to T^{-}}f_{1}(x)\leq nf_{1}(k \frac{T}{n}) A = lim ⁡ x → 0 + f 1 ( x ) ≤ n f 1 ( k T n ) and B = lim ⁡ x → T − f 1 ( x ) ≤ n f 1 ( k T n )

for every

k=1,2,...,n-1, (n\geq 2) k = 1 , 2 , . . . , n − 1 , ( n ≥ 2 )

.