  It is well-known that gravitationally induced vacuum polarization often
violates the point-wise energy conditions and sometimes violates the averaged
energy conditions. In this paper I begin a systematic attack on the question of
where and by how much the various energy conditions are violated. I work in the
test-field limit, and focus on conformally coupled massless scalar fields in
Schwarzschild spacetime, using the Hartle--Hawking vacuum. I invoke a mixture
of analytical and numerical techniques, and critically compare the qualitative
behaviour to be expected from the Page approximation with that adduced from the
numerical calculations of Anderson, Hiscock, and Samuel. I show that the
various point-wise energy conditions are violated in a series of onion-like
layers located between the unstable photon orbit and the event horizon, the
sequence of violations being DEC, WEC, and (NEC+SEC). Furthermore the ANEC is
violated for *some* of the null geodesics trapped in this region. Having
established the basic machinery in this paper, the Boulware vacuum will be
treated in a companion paper, while other exensions should be straightforward.
