Magic Mirrors, Dense Diameters, Baire Category

An old result of Zamfirescu says that for most convex curves C in the plane most points in R2 lie on infinitely many normals to C, where most is meant in Baire category sense. We strengthen this result by showing that 'infinitely many' can be replaced by 'continuum many' in the statement. We present further theorems in the same spirit.