  An N-parameter Brownian sheet in R^d maps a non-random compact set F in R^N_+
to the random compact set B(F) in \R^d. We prove two results on the image-set
B(F):
  (1) It has positive d-dimensional Lebesgue measure if and only if F has
positive (d/2)-dimensional capacity. This generalizes greatly the earlier works
of J. Hawkes (1977), J.-P. Kahane (1985a; 1985b), and one of the present
authors (1999).
  (2) If the Hausdorff dimension of F is strictly greater than (d/2), then with
probability one, we can find a finite number of points \zeta_1,...,\zeta_m such
that for any rotation matrix \theta that leaves F in B(\theta F), one of the
\zeta_i's is interior to B(\theta F). In particular, B(F) has interior-points
a.s. This verifies a conjecture of T. S. Mountford (1989).
  This paper contains two novel ideas: To prove (1), we introduce and analyze a
family of bridged sheets. Item (2) is proved by developing a notion of
``sectorial local-non-determinism (LND).'' Both ideas may be of independent
interest.
  We showcase sectorial LND further by exhibiting some arithmetic properties of
standard Brownian motion; this completes the work initiated by Mountford
(1988).

Probability