After PCA, will the points on the convex hull still be on the hull in the low-dimension space?

Question

I have thousands of spectral reflectance data with dimension of 31, in order to reduce the data size to operate some clustering algorithms, I have to use PCA to reduce the dimension of spectral reflectance matrix. So my question is, if some points in the 31-dimension hyperspace is the vertices of the convex hull, will these points still be the vertices in the low-dimension hyperspace after PCA? My intuitive answer is YES, but is there any paper gave the proof? Many thanks.