Examples

Parallel X-ray of a convex body. The parallel X-ray \(X_uK\) of a convex body \(K\) in R\(^n\) in the direction \(u\in S^{n-1}\) is the function giving for each \(x\in u^{\perp}\) the length of the intersection of \(K\) with the line through \(x\) parallel to \(u\). In short, \(X_uK\) gives the lengths of all the chords of \(K\) that are parallel to \(u\).

 Parallel X-ray of a convex body KFigure 1: Parallel X-ray of a convex body K

Point or fan-beam X-ray of a convex body. The point X-ray \(X_pK\) of a convex body \(K\) at a point \(p\) in R\(^n\) is the function giving for each \(u\in S^{n-1}\) the length of the intersection of \(K\) with the line through \(p\) parallel to \(u\). In short, \(X_pK\) gives the lengths of all the chords of \(K\) contained in lines through \(p\).

Point X-Rays on a Convex Body Figure 2: Possible examples of pairs of different convex bodies with equal X-rays at two points. It is an open problem whether such pairs actually exist.

Discrete X-ray of a finite set. The discrete X-ray of a finite subset \(E\) of R\(^n\) in the direction \(u\in S^{n-1}\) is the function giving for each \(x\in u^{\perp}\) the cardinality of the intersection of \(E\) with the line through \(x\) parallel to \(u\). In short, it gives the number of points in \(E\) lying on each line parallel to \(u\).

Example of Discrete X-Ray Figure 3. Different convex lattice sets with equal discrete X-rays in each of the 6 directions shown. One set contains the black and half-black points and the other contains the grey and half-grey points.

Projection function of a convex body. Suppose that \(i\in \{1,\dots,n-1\}\). The \(i\)th projection function of a convex body \(K\) in R\(^n\) gives for each \(i\)-dimensional subspace \(S\) in R\(^n\) the \(i\)-dimensional volume \(V_i(K|S)\) of the projection of \(K\) onto \(S\). In particular, \(i=1\) corresponds to the width function that gives the width of \(K\) in all directions and \(i=n-1\) corresponds to the brightness function that gives the areas (\((n-1)\)-dimensional volumes) of the projections of \(K\) onto hyperplanes.

Brightness Function Figure 4. Brightness function of a tetrahedron records the area, but not the shape, of the shadows on hyperplanes.

Section function of a star body. Suppose that \(i\in \{1,\dots,n-1\}\). The \(i\)th section function of a star body \(K\) in R\(^n\) gives for each \(i\)-dimensional subspace \(S\) in R\(^n\) the \(i\)-dimensional volume \(V_i(K\cap S)\) of the intersection of \(K\) and \(S\). If \(i=n-1\), we simply refer to the section function of \(K\).

Here a star body \(K\) is a set containing the origin and star-shaped with respect to the origin, such that the function giving for each \(u\in S^{n-1}\) the distance from \(o\) to the boundary of \(K\) (called the radial function of \(K\)) is continuous. (Other definitions of the term have been used.)

Figure 5. Radial function \(\rho_K\) of a star body K.