  We give an explicit formula for the probability that the convex hull of an
$n$-step random walk in $R^d$ does not contain the origin, under the assumption
that the distribution of increments of the walk is centrally symmetric and puts
no mass on affine hyperplanes. This extends the formula by Sparre Andersen
(1949) for the probability that such random walk in dimension one stays
positive. Our result is distribution-free, that is, the probability does not
depend on the distribution of increments.
  This probabilistic problem is shown to be equivalent to either of the two
geometric ones: 1) Find the number of Weyl chambers of type $B_n$ intersected
by a generic linear subspace of $R^n$ of codimension $d$; 2) Find the conic
intrinsic volumes of a Weyl chamber of type $B_n$. We solve the first geometric
problem using the theory of hyperplane arrangements. A by-product of our method
is a new simple proof of the general formula by Klivans and Swartz (2011)
relating the coefficients of the characteristic polynomial of a linear
hyperplane arrangement to the conic intrinsic volumes of the chambers
constituting its complement.
  We obtain analogous distribution-free results for Weyl chambers of type
$A_{n-1}$ (yielding the probability of absorption of the origin by the convex
hull of a generic random walk bridge), type $D_n$, and direct products of Weyl
chambers (yielding the absorption probability for the joint convex hull of
several random walks or bridges). The simplest case of products of the form
$B_1\times \dots \times B_1$ recovers the Wendel formula (1962) for the
probability that the convex hull of an i.i.d. multidimensional sample chosen
from a centrally symmetric distribution does not contain the origin.
  We also give an asymptotic analysis of the obtained absorption probabilities
as $n \to \infty$, in both cases of fixed and increasing dimension $d$.
