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We prove the Poincaré inequality for vector fields on the balls of the control distance by integrating along subunit paths. Our method requires that the balls are representable by means of suitable “controllable almost exponential maps”.

Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R^n containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R^2 and R^3. Some new sharp inequalities characterizing parallelotopes in Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided.

We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge-Kutta DG (RKDG) methods.

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