  A classical theorem of Spencer shows that any set system with n sets and n
elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of
Bansal, Lovett and Meka shows that such colorings can be found in polynomial
time. In fact, the Lovett-Meka algorithm finds a half integral point in any
"large enough" polytope. However, their algorithm crucially relies on the facet
structure and does not apply to general convex sets.
  We show that for any symmetric convex set K with measure at least
exp(-n/500), the following algorithm finds a point y in K \cap [-1,1]^n with
Omega(n) coordinates in {-1,+1}: (1) take a random Gaussian vector x; (2)
compute the point y in K \cap [-1,1]^n that is closest to x. (3) return y.
  This provides another truly constructive proof of Spencer's theorem and the
first constructive proof of a Theorem of Gluskin and Giannopoulos.
