$d$-dimensional convex set in $\Bbb{R}^n$

Question

Given a $d$-dimensional convex set $C\subset R^n$, is it possible to translate and rotate $C$ so that the coordinates of the points in $C$ is $0$ for $n-d$ axes of $\Bbb{R}^n$?

Intuitively, I believe this to be true. Can it be proved in just a few steps?

Answer 1

A convex set has the same dimension as its affine span. So $C$ is contained in a $d$-dimensional affine subspace. A translation takes that to a $d$-dimensional linear subspace. Take an orthonormal basis of the linear subspace, and an orthogonal transformation will take the basis elements to the first $d$ standard unit vectors.