In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of them in half (with respect to their measure, e.g. volume) with a single (n − 1)-dimensional hyperplane. This is possible even if the objects overlap.

It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without stating the theorem in the n-dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey.

The ham sandwich theorem takes its name from the case when n = 3 and the three objects to be bisected are the ingredients of a ham sandwich. Sources differ on whether these three ingredients are two slices of bread and a piece of ham (Peters 1981), bread and cheese and ham (Cairns 1963), or bread and butter and ham (Dubins & Spanier 1961). In two dimensions, the theorem is known as the pancake theorem to refer to the flat nature of the two objects to be bisected by a line (Cairns 1963).

According to Beyer & Zardecki (2004), the earliest known paper about the ham sandwich theorem, specifically the n = 3 case of bisecting three solids with a plane, is a 1938 note in a Polish mathematics journal (Editors 1938). Beyer and Zardecki's paper includes a translation of this note, which attributes the posing of the problem to Hugo Steinhaus, and credits Stefan Banach as the first to solve the problem, by a reduction to the Borsuk–Ulam theorem. The note poses the problem in two ways: first, formally, as "Is it always possible to bisect three solids, arbitrarily located, with the aid of an appropriate plane?" and second, informally, as "Can we place a piece of ham under a meat cutter so that meat, bone, and fat are cut in halves?" The note then offers a proof of the theorem.

A more modern reference is Stone & Tukey (1942), which is the basis of the name "Stone–Tukey theorem". This paper proves the n-dimensional version of the theorem in a more general setting involving measures. The paper attributes the n = 3 case to Stanislaw Ulam, based on information from a referee; but Beyer & Zardecki (2004) claim that this is incorrect, given the note mentioned above, although "Ulam did make a fundamental contribution in proposing" the Borsuk–Ulam theorem.

The two-dimensional variant of the theorem (also known as the pancake theorem) can be proved by an argument which appears in the fair cake-cutting literature (see e.g. Robertson–Webb rotating-knife procedure).

For each angle ${\displaystyle \alpha \in [0,180^{\circ }]}$, a straight line ("knife") of angle ${\displaystyle \alpha }$ can bisect pancake #1. To see this, translate along its normal a straight line of angle ${\displaystyle \alpha }$ from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$; the fraction of pancake #1 covered by the line changes continuously from 0 to 1, so by the intermediate value theorem it must be equal to 1/2 somewhere along the way. It is possible that an entire range of translations of our line yield a fraction of 1/2; in this case, it is a canonical choice to pick the middle one of all such translations.

When the knife is at angle 0, it also cuts pancake #2, but the pieces are probably unequal (if we are lucky and the pieces are equal, we are done). Define the 'positive' side of the knife as the side in which the fraction of pancake #2 is larger. We now turn the knife, and translate it as described above. When the angle is ${\displaystyle \alpha }$, define ${\displaystyle p(\alpha )}$ as the fraction of pancake #2 at the positive side of the knife. Initially ${\displaystyle p(0)>1/2}$. The function ${\displaystyle p}$ is continuous, since small changes in the angle lead to small changes in the position of the knife.