  The notion of an apparent horizon (AH) in a collapsing object can be carried
over from the Lema\^{\i}tre -- Tolman (L--T) to the quasispherical Szekeres
models in three ways: 1. Literally by the definition -- the AH is the boundary
of the region, in which every bundle of null geodesics has negative expansion
scalar. 2. As the locus, at which null lines that are as nearly radial as
possible are turned toward decreasing areal radius $R$. These lines are in
general nongeodesic. The name "absolute apparent horizon" (AAH) is proposed for
this locus. 3. As the boundary of a region, where null \textit{geodesics} are
turned toward decreasing $R$. The name "light collapse region" (LCR) is
proposed for this region (which is 3-dimensional in every space of constant
$t$); its boundary coincides with the AAH. The AH and AAH coincide in the L--T
models. In the quasispherical Szekeres models, the AH is different from (but
not disjoint with) the AAH. Properties of the AAH and LCR are investigated, and
the relations between the AAH and the AH are illustrated with diagrams using an
explicit example of a Szekeres metric. It turns out that an observer who is
already within the AH is, for some time, not yet within the AAH. Nevertheless,
no light signal can be sent through the AH from the inside. The analogue of the
AAH for massive particles is also considered.
