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What's the best way to make the two red corners sharp on this mesh?

My solution so far has been to make it much higher poly, and move some verts to sharpen them, but I want a lower poly solution.

enter image description here

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    $\begingroup$ Never ask for 'best', because that will get your question closed because the answers are opinions. As for 'how would you' and see what turns up, perhaps. $\endgroup$ Oct 19 at 15:55
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My approach that's sometimes useful:

  • Bevel the vertex. That leaves geometry like this:

beveled vertex

  • Select these two points:

selection

  • Merge at either first or last depending on how you want your edge flow.

merging the selection

By itself this doesn't give you a precise corner, but it does allow you to control the curvature of the corner, by edge sliding one of the vertices:

Edge slide to control sharpness

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Something like that doesn't give too much pinching, you need to shrinkwrap it to a high-subdivided cylinder so that it keeps a good circular shape, and apply the Shrinkwrap:

enter image description here

enter image description here

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