  We give the following positive answer to Gromov's question (in "Oka's
principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2,
851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a
compact convex set in a complex Euclidean space C^n to a certain complex
manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the
parametric Oka property. In particular, for any reduced Stein space X the
inclusion of the space of holomorphic maps of X to Y into the space of
continuous maps is a weak homotopy equivalence. This shows that all Oka type
properties of a complex manifold are equivalent to each other. (See also the
articles F. Forstneric, "Runge approximation on convex sets implies Oka's
property", Ann. Math. (2), 163, 689-707 (2006); "Extending holomorphic mappings
from subvarieties in Stein manifolds", Ann. Inst. Fourier 55, 733-751 (2005).)
