English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$and the integration of $h$ on this sublevel set when $g$ and $h$are positivelyhomogeneous functions. For instance, the latter integral reduces to integrating$h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ isa polynomial, then the volume of the sublevel set is a convex function of thecoefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int\phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions$\phi:R_+\to R$. We also provide a numerical approximation scheme to computethe volume or integrate $h$ (or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set $\{x\,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a(hard) convex optimization problem for which we also propose two convergentnumerical schemes. Finally, we provide a Gaussian-like property of non Gaussianintegrals for homogeneous polynomials that are sums of squares and criticalpoints of a specific function.

Level sets and non Gaussian integrals of positively homogeneous functions