  We formulate the conditional Kolmogorov complexity of x given y at precision
r, where x and y are points in Euclidean spaces and r is a natural number. We
demonstrate the utility of this notion in two ways.
  1. We prove a point-to-set principle that enables one to use the
(relativized, constructive) dimension of a single point in a set E in a
Euclidean space to establish a lower bound on the (classical) Hausdorff
dimension of E. We then use this principle, together with conditional
Kolmogorov complexity in Euclidean spaces, to give a new proof of the known,
two-dimensional case of the Kakeya conjecture. This theorem of geometric
measure theory, proved by Davies in 1971, says that every plane set containing
a unit line segment in every direction has Hausdorff dimension 2.
  2. We use conditional Kolmogorov complexity in Euclidean spaces to develop
the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y,
where x and y are points in Euclidean spaces. Intuitively these are the lower
and upper asymptotic algorithmic information densities of x conditioned on the
information in y. We prove that these conditional dimensions are robust and
that they have the correct information-theoretic relationships with the
well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y)
and Mdim(x:y).
