Let f be a nonnegative concave function on a convex domain D. Assume that f achieves a positive value at some point x in D. Then f(y)>0 for all y in D^o.

Trace Glass

Trace Glass

Answered question

2022-10-29

Let f be a nonnegative concave function on a convex domain D. Assume that f achieves a positive value at some point x D. Then f ( y ) > 0 for all y D o .

Answer & Explanation

giosgi5

giosgi5

Beginner2022-10-30Added 15 answers

Hint: Let x be a point in D for which f ( x ) > 0, and suppose (for the purpose of contradiction) that y D o is such that f ( y ) = 0. Note that for t 1, we have
f ( ( 1 t ) x + t y ) ( 1 t ) f ( x ) + t f ( y ) .

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