Intrinsic Entropies of Log-Concave Distributions

The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable X, the sequence of the [n
^θ
]
^th
intrinsic volumes of the typical sets of X in dimensions n ≥ 1 grows exponentially with a well-defined rate. We denote this rate by h
_X
(θ), and call it the θ
^th
intrinsic entropy of X. We show that hX (θ) is a continuous function of θ over the range [0, 1], thereby providing a smooth interpolation between the values 0 and h(X) at the endpoints 0 and 1, respectively.