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I am trying to get my head around convex analysis proofs and I am not sure how to begin.

I think the definition I should use for proofs is that a set C is convex if for any $u,v\in C$, the point $tu+(1-t)v \in C \forall t \in [0,1]$

For example, How would I prove or disprove that the set of matrices containing all even numbers along the diagonal is convex or not?

Rodrigo de Azevedo
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123123
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    Your approach (second paragraph) is correct. I suggest you try applying it to the set described as "matrices containing all even numbers along the diagonal". It is often sufficient to consider just the midpoint between $u,v$, i.e. the case $t=1/2$. – hardmath Aug 02 '20 at 14:41
  • Your definition of a convex set needs to include "for all $t \in [0,1]$." It is not correct as you state. For example, you don't say anything about what $t$ is, perhaps $t=42$? – Michael Aug 02 '20 at 15:22
  • I see you fixed the definition. Now, can you think of two even numbers? What are they?... – Michael Aug 02 '20 at 15:31
  • there is no need for the sarcasm, I wanted validation as to if the definition was the correct approach. – 123123 Aug 02 '20 at 15:48
  • Just start by writing such generic matrices $X, Y$ that have even numbers on the diagonal. What happens if we multiply an even number by some $t_1\neq\frac{1}{2}$, then another even number by $1-t_1$? Will the sum be even? Generally, to "prove" something, you need it to hold for all possible cases. But in order to show it's not true, one count example is enough. – iarbel84 Aug 02 '20 at 15:56
  • @123123 : What sarcasm? I was planning to give you a direct next-step based on the two numbers that you provided. Yet, you never provided them... The next-step would be solidified if you see that it was based on the two number that you chose. – Michael Aug 02 '20 at 16:18

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