We prove that the largest convex shape that can be placed inside a given convex shape $$Q \subset \mathbb {R}^{d}$$ Q ⊂ R d in any desired orientation is the largest inscribed ball of Q . The statement is true both when “largest” means “largest volume” and when it means “largest surface area”. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

Abstract We prove a generalization of Pál's conjecture from 1921: if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously… Expand

where Sn−1 is the unit sphere in R. This paper will be mainly concerned with the following issue, which is still poorly understood: what metric restrictions does the property (1) put on the set E?… Expand

Helps students see mathematics as an organic whole by focusing on the geometric while presenting viewpoints and methods that require a general understanding and unification of previous mathematical… Expand

We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic… Expand

Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably… Expand