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 $u$.

Figure 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 $p$.

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 $u$.

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.

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.)