Determining indices upon which a sequence of real numbers is convex

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.

Answer 1

[Roger Stafford]

k1 = k+1-find(diff([x(k+1:-1:1),-inf],2)<=0,'first');
k2 = k-1+find(diff([x(k-1:n),-inf],2)<=0,'first');

[Paul Fishback]

Thanks Roger!