Rotational Crofton formulae for flagged intrinsic volumes

By Jérémy Auneau-Cognacq
PhD Dissertations
October 2010
Abstract:

The study of stereology can be traced back to the 18th century with the celebrated Buffon's needle problem, which was posed for the first time by the Comte de Buffon and can be described as a method for estimating $\pi$ by throwing needles on a parquet floor. More generally, stereology is the study of intrinsic geometrical properties of a set through measurements made on lower dimensional sections of that set. For practical reasons, we may require that those sections go through a fixed point in space, instead of being randomly positioned. The study of geometrical properties in that particular set-up is called local stereology, the foundations of which were laid by Eva B. Vedel Jensen.

The focus of interest of the present thesis is on integral geometric identities of the type

$\beta(X) = \int \alpha(X \cap L) \, \mathrm{d} L$
where $\alpha$ and $\beta$ are geometrical quantities of a set $X$, such as its volume, surface area or, more generally, intrinsic volume, and the integration is over all sections containing the fixed point origo. Our main result is a local stereological analogue to the well-known Crofton formula. More precisely, we derive geometric formulae that relate new flagged intrinsic volumes of a set $X$ with the flagged intrinsic volumes of its sections, $X \cap L$. The development of a potential local stereological analogue of the famous principal kinematic formula is also discussed.

Thesis advisor: Eva B. Vedel Jensen
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