  We prove a transversality "lifting property" for compactified configuration
spaces as an application of the multijet transversality theorem: the
submanifold of configurations of points on an arbitrary submanifold of
Euclidean space may be made transverse to any submanifold of the configuration
space of points in Euclidean space by an arbitrarily $C^1$-small variation of
the initial submanifold, as long as the two submanifolds of compactified
configuration space are boundary-disjoint. We use this setup to provide
attractive proofs of the existence of a number of "special inscribed
configurations" inside families of spheres embedded in $\mathbb{R}^n$ using
differential topology. For instance, there is a $C^1$-dense family of smooth
embedded circles in the plane where each simple closed curve has an odd number
of inscribed squares, and there is a $C^1$-dense family of smooth embedded
$(n-1)$-spheres in $\mathbb{R}^n$ where each sphere has a family of inscribed
regular $n$-simplices with the homology of $O(n)$.
