Lp-norms, Log-barriers and Cramer transform in Optimization

We show that the Laplace approximation of a supremum by Lp-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate. In addition, minimizing the LBF of the dual is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem in cases when the Legendre-Fenchel conjugate of g cannot be derived explicitly from its definition

The Law of the Iterated Logarithm for Lp-Norms of Kernel Estimators of Cumulative Distribution Functions

In this paper, we consider the strong convergence of Lp-norms (p ≥ 1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF

Lp-norms of polynomials with positive real part

AbstractWe derive an estimate for Δn, 1 = sup{(2π)−1 ∝02π¦p(eit)¦dt: p(z) = 1 + a1z + · · · + anzn, Re(p(z)) > 0 for ¦z¦ < 1}. In particular it is shown that Δn, 1 ⩽ 1 + log(C1(n + 1) + 1), where C1 = 0.686981293…, It is also shown that 2π ⩽ lim infn → ∞ Δn, 1log n. Finally, upper bounds are found for the Lp-norms of polynomials with positive real part on the unit disk