• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.3
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• pp.37-41
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• 2007

The semi-convexity for planar shapes has been recently introduced in [2]. As a generalization of the convextiy, semi-convexity is closed under the Minkowski sum. But the definition of semi-convexity requires that the shape boundary should satifisfy a differentiability condition $C^{1:1}$, which means that it should be possible to take the normal vector field along the domain's extended boundary. In view of the fact that the semi-convextiy is a most natural generalization of the convexity in many respects, this is a severe restriction for the semi-convexity, since the convexity requires no such a priori differentiability condition. In this paper, we generalize the semi-convexity to the closure of the class of semi-convex $\mathcal{M}$-domains for any Minkowski class $\mathcal{M}$, and show that this generalized semi-convexity is also closed under Minkowski sum.