Suppose is a bounded domain and is . If there exists a **convex** function satisfies

Then is uniformly convex. In other words, the principle curvature of every point on , namely , are positive. Moreover, is connect.

For any boundary point of , we may suppose the point is origin. Since , there exists a neighborhood of 0 such that is represented by with . Choosing the principle coordinate system, poositive axis is interior normal at origin and

here .

Since near the origin, it following, on differentiation,

for .

So at origin, we get

which means

So on and for all . However, is continuous function on , which means does not change sign on .

Since is convex and on , then in . Also we know that is positive definite matrix. Thus in , by the hopf lemma,

at

So near origin is almost , which is positive by the property of . So is positive.

For the argument is connect, see the paper in the following remark.

Caffarellli, Nirenberg and Spruck.* The Dirichlet problem for nonlinear second order elliptic equations,III:Functions of the eigenvalues of the Hessian*.

Also refer to the formula [GT, p471].

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