  Let $S_k$ be a random walk in $R^d$ such that its distribution of increments
does not assign mass to hyperplanes. We study the probability $p_n$ that the
convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does
not include the origin. By providing an explicit formula, we show that for
planar symmetrically distributed random walks, $p_n$ does not depend on the
distribution of increments. This extends the well known result by Sparre
Andersen (1949) that a one-dimensional random walk satisfying the above
continuity and symmetry assumptions stays positive with a distribution-free
probability. We also find the asymptotics of $p_n$ as $n \to \infty$ for any
planar random walk with zero mean square-integrable increments.
  We further developed our approach from the planar case to study a wide class
of geometric characteristics of convex hulls of random walks in any dimension
$d \ge 2$. In particular, we give formulas for the expected value of the number
of faces, the volume, the surface area, and other intrinsic volumes, including
the following multidimensional generalization of the Spitzer--Widom formula
(1961) on the perimeter of planar walks: $$ E V_1 (conv(0, S_1, \dots, S_n)) =
\sum_{k=1}^n \frac{E \|S_k\|}{k}, $$ where $V_1$ denotes the first intrinsic
volume, which is proportional to the mean width.
  These results have applications to geometry, and in particular, imply the
formula by Gao and Vitale (2001) for the intrinsic volumes of special
path-simplexes, called canonical orthoschemes, which are finite-dimensional
approximations of the closed convex hull of a Wiener spiral. Moreover, there is
a direct connection between spherical intrinsic volumes of these simplexes and
the probabilities $p_n$.
  We also prove similar results for convex hulls of random walk bridges, and
more generally, for partial sums of exchangeable random vectors.
