This talk is concerned with the probabilistic behaviour of a condition
number C(A) for the problem of deciding whether a system of n
homogeneous linear inequalities in m unknowns has a non-zero solution.
In the case where the input system is feasible, the exact probability
distribution of the condition number for random inputs is determined,
and a sharp bound for the general case. In particular, for the
expected value of the logarithm log C(A), an upper bound of order
O(log m) (m the number of variables) is presented which does not
depend on the number of inequalities.
The probability distribution of the condition number C(A) is closely
related to the probability of covering the m-sphere with n spherical
caps of a given radius. As a corollary, we obtain bounds on the
probability of covering the sphere with random caps.