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I find a proof here
http://mathhelpforum.com/advanced-math-topics/129503-second-order-condition-convexity.html
But I don't understand the second part. For example.
What does

For the converse, if f is convex, then certain subdeterminants of the Hessian matrix satisfy certain conditions.

mean?

can someone help me?

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Are you unfamiliar with some of the terms? Are you not sure what conditions are being talked about? Where is it you are stuck? – Alex Becker♦ Jun 11 '12 at 6:57
Yes, maybe. I don't know what the two 'certain' in the sentence mean.@AlexBecker – cloudygoose Jun 11 '12 at 7:26

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