An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on Green and Tao's work on ordinary lines in the plane, combined with classical results on space quartic curves and non-generic projections of curves. This gives an alternative approach to Ball's recent results on ordinary planes, as well as extending them. We also give bounds on the number of coplanar quadruples determined by a finite set of points on a rational space quartic curve in complex 3-space, answering a question of

Let E n denote the (real) n-dimensional Euclidean space. It is not known whether an equilateral set in the 1 sum of E a and E b , denoted here as E a ⊕ 1 E b , has maximum size at least dim(E a ⊕ 1 E b) + 1 = a + b + 1 for all pairs of a and b. We show, via some explicit constructions of equilateral sets, that this holds for all a 27, as well as some other instances.

Let E n denote the (real) n-dimensional Euclidean space. It is not known whether an equilateral set in the ℓ 1 sum of E a and E b , denoted here as E a ⊕ 1 E b , has maximum size at least dim(E a ⊕ 1 E b ) + 1 = a + b + 1 for all pairs of a and b. We show, via some explicit constructions of equilateral sets, that this holds for all a 27, as well as some other instances.

Let P be a set of n points in real projective d-space, not all contained in a hyperplane, such that any d points span a hyperplane. An ordinary hyperplane of P is a hyperplane containing exactly d points of P. We show that if d 4, the number of ordinaryif n is sufficiently large depending on d. This bound is tight, and given d, we can calculate the exact minimum number for sufficiently large n. This is a consequence of a structure theorem for sets with few ordinary hyperplanes: For any d 4 and K > 0, if n C d K 8 for some constant C d > 0 depending on d, and P spans at most K n−1 d−1 ordinary hyperplanes, then all but at most O d (K) points of P lie on a hyperplane, an elliptic normal curve, or a rational acnodal curve. We also find the maximum number of (d + 1)-point hyperplanes, solving a d-dimensional analogue of the orchard problem. Our proofs rely on Green and Tao's results on ordinary lines, our earlier work on the 3-dimensional case, as well as results from classical algebraic geometry.