I'm curious to understand in several manner, what is the metric geometry of the metric space homeomorphic to a sphere, obtained from a compact convex set $K\subset R^2$ with the Euclidean distance, when one makes the following quotient that the border $\partial K$ is one point, say $p$.

The space is certainly geodesic and even locally Euclidean except in a neighborhood of $p$. The space of directions around this point seems to have an infinite angle.

Can the space be represented is some nice manner as a (non-convex) surface of $R^3$? What happens if $K$ is a disk? Or $K$ is a square?