Can a part of the spherical surface be convex?

Question

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

Answer 1

"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