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In my opinion, the line between two arbitrary points on the surface of a sphere is never part of the surface (the line is inside of the sphere). Hence a part of the spherical surface can't be convex.

But I have read it differently.

E.g. here: https://www.jstor.org/stable/1969084?seq=1

or here: https://projecteuclid.org/download/pdf_1/euclid.bams/1183500307

Ethan Bolker
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Dex124
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    Please tell us where you've read this. It's impossible to address your question without that context. – Brian Borchers Jul 23 '20 at 13:15
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    Different things can be meant by "convex surface" vs "convex set". Compare to the use of "convex" for the graph of a concave up function. You might find the discussion on this question helpful: https://math.stackexchange.com/questions/623728/the-definition-of-convex-surface – halrankard Jul 23 '20 at 13:17
  • Thanks @BrianBorchers I added two links – Dex124 Jul 23 '20 at 13:41
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    Thanks @halrankard thats it :) – Dex124 Jul 23 '20 at 13:46

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"Convex on the surface of the sphere" means convex with respect to geodesics (great circles) on the sphere.

From the first paragraph of the linked article:

By a convex region on the sphere we mean a region such that any great circle arc of length less than $180°$, whose end points lie in the region, lies entirely in the region

Ethan Bolker
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