## Warm-up

Let's start in the plane.

A curve of constant width can be thought of in terms of a slab in the plane of the given width that moves around as a function of angle. A moving slab has an envelope, which is the curved form by the points where the boundary lines are instantaneously rotating (*i.e.*, the limit of where nearby points cross). The pair of curves touch the slab at the ends of a perpendicular to the slab. The midpoint of this perpendicular segment moves forward or backward along the axis of the slab.

To put it another way: the (3-dimensional) tangent line bundle to the plane has a **contact structure**, which is a 2-plane field giving directions of "allowed" motions. It's like an ice skate: it can move forward or backward, and it can rotate. A curve following these rules is called a Legendrian curve, and curves of constant width correspond to Legendrian sections of the fibration (tangent line bundle of plane) --> (sets of parallel planes = $\mathbb {RP}^1$.

The Legendrian sections also correspond to differentiable curves in the plane, generically with an odd number of cusps (unless the singularities are degenerate), whose tangent line turns consistently by 180 degrees in one direction. For any such curve and any $w$, draw the figure swept out by the perpendiculars in each direction a distance $w$: it is a curve of constant width.

## Three dimensions

Added: See Ivanov's answer for a more efficient route for deducing it's a sphere.

If every projection of a surface $FS$ (fake sphere) in 3 dimensions is a circle, then we can think about the family of cylinders that enclose it. This is codified as a map from $\mathbb {RP}^2 = $ the set of parallelism classes of lines
to the set of lines in $\mathbb R^3$ that is a section of the parallelism equivalence relation.

From the two-dimensional picture, we can visualize what happens in a slice:
For each tangent direction to $\mathbb {RP}^2$, there is an axis about which the cylinder is instantaneously rotating. The common perpendicular to the axis of rotation and the axis of the cylinder intersects the cylinder at two points where the surface is tangent to the cylinder. This line segment is inside $FS$.

Now think about a different tangent direction to the same point of $\mathbb {RP}^2$. We get another axis of rotation, describing what happens in that direction. But, if the new line segment did not intersect the first, the first line segment would escape from the moving cylinder.

We conclude that all axes for any point in $\mathbb{RP}^2$ coincide, and $FS$ contains a round disk passing through the common point of these axes. This defines a map from $\mathbb {RP}^2$ to the tangent line bundle of $\mathbb{R}^3$, where $FS$ is swept out by
circles centered at the tangent lines in the plane perpendicular to the line.

However, the base point for the tangent line must be constant, because if it were to move, then the sideways view would contain a constant width strip about the image, and this is not contained in a disk unless the map is constant.

Conclusion: $FS$ is not fake. It is a genuine $S^2$. You can go ahead and buy it.

## What can you learn from a set of Shadows?

Suppose we have a collection of shadows of an unseen object that is moving in unknown ways---we don't know the projections corresponding to which shadow. What can we deduce?

I don't have an answer, but here are some things that can be done as a start:

The set of closed subsets in the Euclidean plane of any given bounded diameter has a topology and metric, the Hausdorff topology and metric. (More precisely, we're taking the quotient space of the Hausdorff topology by a compact equivalence relation, and using minimum Haudorff distance between equivalence classes to induce a metric on the quotient.)
The profile of a projection is a smooth function of the projection, so we get a also get a smooth structure on this space: a diffeomorphism to $\mathbb{RP}^2$.

Let's think first about a generic, smooth convex shape, for which every projection is different, so the set of our shadows is homeomorphic to $\mathbb{RP}^2$. We can try to reconstruct successively more information: the projective structure, the metric structure, and finally the set of solid cylinders in space which enclose the shape.

A line, in the projective structure, consists of a set of profiles that are perpendicular projections to a collection of planes that share a line.

For any profile, there is a circle's worth of 1-dimensional projection, with an invariant, the width. Critical points of width are projections for which the line between points mapping to extremes of the projection are perpendicular to the surface. Think of the pair of planes tangent to the surface at the endpoints of such a line segment.

As one moves around in $\mathbb{RP}^2$, the multiset of critical widths changes.
Consider a profile where one of these critical widths is is a critical point with respect to the space of profiles. This happens when the pair of tangent planes that project to tangent lines of the profile are perpendicular to the line segment connecting the points of tangency. At any such point: for instance, when the diameter is maximal or minimal--- we can spin the projection around the axis to get an entire circle's worth of profiles sharing the same critical width.

A better way to think of these globally critical widths is to imagine a pair parallel planes squeezing down on the surface, a kind of caliper. This makes width a function on $\mathbb{RP}^2$.
A Morse function on $\mathbb{RP}^2$ has at least 3 critical points, so there are at least 3 globally critical widths.

In this way, we get an initial network of lines for the projective structure we're seeking.
Any two lines in $\mathbb{RP}^2$ intersect. Furthermore, we know the angles between these projective lines, since any two lines intersect, and in the profile corresponding to the intersection, we see two diameters at once with angle equal to their angle in space.

Once we have a projective line identified together with its axis, we can deduce the profile in the projection that maps the axis to a point, up to diffeomorphisms of $\mathbb R^2 \setminus 0$ that take lines to lines and act as rotations on any one line. The information, in other words, is a curve in the positive orthant that tells the pair of distances of intersection points of the curve with lines through the origin. Generically, there is only one profile in our collection that matches, so we can deduce what it is. This gives the information we need to get the angle parametrization of our projective line.

I think we're well on the way to complete identification of a generic smooth convex shape, but,
I'll leave it here for now. Feel free to add, refine, or streamline..

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