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Determining indices upon which a sequence of real numbers is convex

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I have a vector, X, of real numbers which has a minimum at index k, where 1<k<n.
I want to determine the smallest index k1 and largest index k2, for which the sequence is strictly convex on k1,..,k,...k2. Here, convexity means 2X(i) < X(i-1)+X(i+1) for k1+1<=i<=k2-1.
In calculus terms, this question is analogous to determining the largest subinterval about a local minimum upon which the function's second derivative is positive.

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