While it is unlikely to occur naturally...
A hollow sphere structure is still possible, theoretically.
If the shell of the world is thick enough and strong enough, then it could withstand the stresses of rotation, and possibly gravitational stresses of surrounding celestial objects. This is also the same basic structure of a Dyson Sphere.
Gravity is produced by mass, it doesn't matter what shape it is in. Therefore, a solid sphere of sufficient mass will produce gravity from the center. A giant planet sized pyramid of mass would produce gravity from the center, which would be feel stronger on the planes (closer to the center) and weaker on or near the points. A giant cube of mass would have a similar effect. Things get a bit strange with a hollow object.
A hollow sphere will produce gravity not from the center of the sphere, but from the center of the existing mass, which in this case is material making up the shell of our hollow sphere. For the sake of theory, let's say our hollow shell-sphere is the size of Jupiter, with no mass at all in the center. The thickness of the sphere let's set at the same as the diameter of the earth. If gravity has the same effects, then what would happen is that the middle part of the shell would melt, and eventually compress to metal as the heavier elements precipitate or percolate out of the molten material.
Note that this would happen everywhere throughout the shell-sphere. The theoretical end result would be a shell which is comprised of an inner layer of rock, under which is a layer of mantle, under which is a layer of liquid rock, and finally a "core" of metal, only this would actually be a sheet of metal. On the other side of the sheet of metal would be another set of layers, liquid rock, mantle, and then outer surface of rock.
This would then produce 1g across the entirety of both inside and outside surfaces of the sphere (assuming, of course, that this structure was actually somehow strong enough to retain its shape). However a sphere like this is most likely to be a complete sphere, with only small openings, unlike your picture.
Your picture will not likely work, based on science...
Gravity on the object in your picture would draw towards the greatest density of mass, in other words, the top and the bottom. This would mean you could stand on the outside or inside of the top or bottom half, as well as stand on the outside or inside of either sphere. Gravity would be less towards the edges, as the amount of mass is also less.
Unless there are external or additional forces not depicted, there is nothing stopping the two halves from exerting gravitational attraction and collapsing into each other. Not to mention the seemingly free-floating islands near the edges of the top and bottom.
But, since you didn't use any "science" tags... it could actually work.
However, as you did not use any of the Science tags, you could postulate magic, handwavium science, other unknown elements, forces, or environments which would in fact support and preserve such a structure as depicted.
A bright idea...
Such a world would be placed under extreme gravitational stress if it were to exist in a solar system by the sun and other planets (unless again, you opt for magic or handwavium). Should it exist in a traditional solar system, then the light of the sun will only reach certain parts of the world, casting much of it into shadow most of the time. While that could be thematically and plot-wise an interesting aspect, it would certainly limit what kinds of life (flora and fauna) could successfully exist and adapt.
If the central pillar-like mountain structure was composed of or covered in a highly reflective material, that would alleviate some of the lighting issue, but would also make day/night cycles rather complicated. Also, note that if the orbital path was oriented so that the open middle part was in the plane of the elliptic - with no axial tilt, then there would not be night in many places of the world, only periods of direct light, and indirect light.