Suppose $f$ is defined on an interval $I$ such that for any compact interval $J \subset I$, and any linear function $L(x) = ax + b$, we have $sup_{J} (f-L) = sup_{bdy(J)} (f-L)$. Is $f$ necessarily convex on $I$?
Equivalent condition for convexity
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real-analysis
convex-analysis
1 Answers
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Yes. Taking $J = [x_0, x_1]$, choose $a$ and $b$ so that $(f-L)(x_0) = (f-L)(x_1)$. Then the condition says that for $0 \le t \le 1$, $$(f-L)(t x_0 + (1-t) x_1) \le t (f-L)(x_0) + (1-t) (f-L)(x_1)$$ and thus $$ f(t x_0 + (1-t) x_1) \le t f(x_0) + (1-t) f(x_1) $$ which is the definition of convexity.