A note on Alexsandrov type theorem for k-convex functions
نویسندگان
چکیده
A classical result of Alexsandrov [1] asserts that convex functions in R are twice differentiable a.e., (see also [3], [8] for more modern treatments). It is well known that Sobolev functions u ∈ W , for p > n/2 are twice differentiable a.e.. The following weaker notion of convexity known as k-convexity was introduced by Trudinger and Wang [12, 13]. Let Ω ⊂ R be an open set and C(Ω) be the class of continuously twice differentiable functions on Ω. For k = 1, 2, . . . , n and a function u ∈ C(Ω), the k-Hessian operator, Fk, is defined by
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