It is shown that if X 1 , X 2 , … X_1,X_2,\ldots are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum \[ Ent ⁡ ( X 1 + ⋯ + X n n ) \operatorname {Ent} \left (\frac {X_{1}+\cdots + X_{n}}{\sqrt {n}} \right ) \] is an increasing function of n n . The result also has a version for non-identically distributed random variables or random vectors.

We study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition.

If K and B are subsets of a vector space (or just a group, or even a homogeneous space), the covering number of K by B, denoted N (K, B), is the minimal number of translates of B needed to cover K. Similarly, the packing number M (K, B) is the maximal number of disjoint translates of B by elements of K. The two concepts are closely related; we have Na ball in a normed space (or in an appropriate invariant metric) and K a subset of that space (the setting and the point of view we will usually employ), these notions reduce to considerations involving the smallest ǫ-nets or the largest ǫ-separated subsets of K.Besides the immediate geometric framework, packing and covering numbers appear naturally in numerous subfields of mathematics, ranging from classical and functional analysis through probability theory and operator theory to information theory and computer science (where a code is typically a packing, while covering numbers quantify the complexity of a set). As with other notions touching on convexity, an important role is played by considerations involving duality. The central problem in this area is the 1972 "duality conjecture for covering numbers" due to Pietsch which has been originally formulated in the operator-theoretic context, but which in the present notation can be stated as Conjecture 1 Do there exist numerical constants a, b ≥ 1 such that for any dimension n and for any two symmetric convex bodies K, B in R n one hasAbove and in what follows A • := {u ∈ R n : sup x∈A x, u ≤ 1} is the polar body of A; "symmetric" is a shorthand for "symmetric with respect to the origin" and, for definiteness, all logarithms are to the base 2. In our preferred setting of a normed space X, the proper generality is achieved by considering log N (K, t B) for t > 0, where B is the unit ball and K -a generic (convex, symmetric) subset of X. The polars should then be thought of as subsets of X * , with B • the unit ball of that space. With minimal care, infinite-dimensional spaces and sets may be likewise considered. To avoid stating boundedness/compactness hypotheses, which are peripheral to the phenomena in question, it is convenient to allow N (•, •), M (•, •) etc. to take the value +∞.The quantity log N (K, t B) has a clear information-theoretic interpretation: it is the complexity of K, measured in bits, at the level of resolution t with respect to the metric for which B is the unit ball. Accordingly, (1) asks whether the complexity of K is controlled by that of the ball in the dual space with respect to • K • (the gauge of K • , i.e., the norm whose unit ball is K • ), and vice versa, at every level of resolution. [The original formulation of the conjecture involved

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