I think not; consider pseudosphere, there is no obvious center. Its embedding in 3D space has infinite dimensions but the object itself has finite surface area.

I think not; consider pseudosphere, there is no obvious center. Its embedding in 3D space has infinite dimensions but the object itself has finite surface area. https://en.wikipedia.org/wiki/Pseudosphere https://en.wikipedia.org/wiki/Pseudosphere If you mean object with finite dimensions and negative curvature and no center, there seems to be such a thing, see Fig. n066200e: https://encyclopediaofmath.org/wiki/Negative_curvature,_surface_of https://encyclopediaofmath.org/wiki/Negative_curvature,_surf... EDIT Rozendorn's idea seems to be still an open question: https://mathoverflow.net/questions/52851/the-geometry-of-nadirashvilis-complete-bounded-negative-curvature-surface https://mathoverflow.net/questions/52851/the-geometry-of-nad...