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This article concerns a class of open billiards consisting of a finite numberof strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ ofthe billiard ball map is a topological Cantor set and its Hausdorff dimensionhas been previously estimated for billiards in $\mathbb{R}^2$, using well-knowntechniques. We extend these estimates to billiards in $\mathbb{R}^n$, and makevarious refinements to the estimates. These refinements also allow improvementsto other results. We also show that in many cases, the non-wandering set isconfined to a particular subset of $\mathbb{R}^n$ formed by the convex hull ofpoints determined by period 2 orbits. This allows more accurate bounds on theconstants used in estimating Hausdorff dimension.

Estimates of Hausdorff dimension for non-wandering sets of higher dimensional open billiards