Title: Support functions, projections and Minkowski addition of Legendrian cycles

Authors: Andreas Bernig

Issue: Volume 55 (2006), Issue 2, 443-464

Abstract: The structure of the space $\mathcal{L}\mathcal{C}(\mathbb{R}^n \times S^{n-1})$ of compactly supported integral Legendrian cycles on $\mathbb{R}^n \times S^{n-1}$ is studied using geometric measure theory. To each such cycle $T$, a support function $h_T \in L^1(S^{n-1}, \mathbb{Z}[\mathbb{R}])$ is constructed. For almost all $k$-dimensional linear subspaces $L$, a Legendrian cycle $\pi_L(T)$, whose support function is the restriction of the support function of $T$ to $L \cap S^{n-1}$, is constructed. It is shown that there is a partial ring structure on $\mathcal{LC}(\mathbb{R}^n \times S^{n-1})$, the multiplication being a generalized Minkowski addition. An example of non-existence of the Minkowski sum is given, but it is shown that the Minkowski sum does exist after changing one of the summands by an arbitrarily small linear map. Finally, mean projection formulas are formulated and proved in the context of Legendrian cycles.