An affine invariant p-th moment measure is defined for a random vector and used to prove sharp moment-entropy inequalities that are more general and stronger than standard moment-entropy inequalities.

A unified approach is presented for establishing a broad class of Cram\'er-Rao inequalities for the location parameter, including, as special cases, the original inequality of Cram\'er and Rao, as well as an $L^p$ version recently established by the authors. The new approach allows for generalized moments and Fisher information measures to be defined by convex functions that are not necessarily homogeneous. In particular, it is shown that associated with any log-concave random variable whose density satisfies certain boundary conditions is a Cram\'er-Rao inequality for which the given log-concave random variable is the extremal. Applications to specific instances are also provided.

Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies.

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in this paper, we construct two new primitive Evans triangles by using the Pell equation x^2-Dy^2=1, and give the two forms of the Evans triangle and the corresponding Evans ratio, we also get a new conclusion that the ratio is an integer from a few different forms of the bottom and high.