Why do you want to make it known? So that you can use it? So that others can use it? Even if it is not for journal publication, is it something you want to have people citing with your name, perhaps in a negative context? The answer to the title question is yes, but there may be career implications and others for which you want answers.

At some point an anologous problem observing 3d objects in 3d space can be considered, but then the steradial projection may vary for other reasons, complicating the analysis. There are enough subtleties in this version that I think it important to consider the vision problem in this special form.

@Joseph, I suspect for the disk it will be a surface of revolution based on a curve simply derived from the graph that Douglas Zare provided to your question. I conjecture ellipsoid just to give people something to refute, even though it is a natural first guess. For a rectangular region, one could imagine a parameterized surface which sliced in one direction resembles circular arcs of measure alpha and sliced in an orthogonal direction has measure beta; I expect reality will be more complicated. In order to draw out the subtleties, I proposed an H region instead.

Why is p needed for the definition?

You might prefer an alternate question even in just 3 dimensions: given a region R and quantity q, say contained in some planar polygon, what is the locus of points in space from which region R subtends a solid angle of measure q?

I don't think I should challenge your reasoning as much as I should challenge my intuition. My main stumbling block is that the diameter that is perpendicular to the radial line from the viewer appears to stay the same length. The pictures Joseph provided do seem to confirm the no answer.

Thank you for lending credance to my earlier thoughts. When I use the flap model, I find the quadrant being partly covered as the flap goes toward the center of the steradial sphere. However, the base of the cone is two such flaps, and the flap moving away produces a smaller steradial projection that goes to zero. I am hoping Joseph will make a picture of the flap model to confirm. I am uncertain about the steradial variation: I am certain that the area does not approach pi steradians.

I am trying a mental simulation where I attach a semicircle to a sphere of same radius, and then (while having the diameter of the semicircle fixed and tangent to the sphere) folding this like a flap and imagining the change in steradial projection. I am not getting a quadrant filling shadow but something else. I am withdrawing my earlier vote.

I think an enlightening picture would be a series that shows the projection onto the steradian sphere as the vertex goes from 90 degrees down to zero. You should see a circle elongate into an ellipse, and almost become but stay within a 1/4 wedge of the steradian sphere.

I vote no to Q1. Imagine the steradian sphere the same diameter as the sphere circumscribing the cone, with the cone rays extended to cut both spheres. As I bring the steradian sphere very close to the equator, I see the steradian increase to close to pi, since the cone vertex approximates the edge of a cube. If you can, do some computation with the vertex at 1 or 0.1 degrees off the horizon/equator.

Consider the adjacency relationships: How many squares share part of a side with a given square. I think you will find that there will be tight restrictions on b and c given that you can't have too many smaller squares surrounding a square of size c.

ADV is absolute determinant value.

It is likely an irrational number between 70/27 and 71/27. Letting T be the tail of the sum n^-n starting with n=4, one has exp(T)70/27 as a tighter upper bound, I think.

On second thought, I may be confusing "totally real" with "formally real". Perhaps a model-theorist can pop in here to help.

The gut feeling is that totally real fields "should have enough continuity" to produce such an element. Is existence enough, or are you hoping for a density result that would say that given z and delta, there is x within delta of z so that the min. poly. of x has the desired property?

@Ioachim, Volume 2 is out? Tell me more! I haven't found it yet.