Shadow Boundaries of Convex Bodies
Shadow Boundaries of Convex Bodies
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凸体的阴影边界
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发表时间：
2013
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通讯作者：
Louise Jottrand
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作者：
Louise Jottrand
如果C是R^n中的凸体，X是R^n的k维线性子空间，我们用S(C，X)表示C在X上的影子边界，它是指属于C的所有点的集合及其与X垂直的切线(n-k)-fl。对于R^3中几乎所有的方向，阴影边界是一条包围身体C的曲线。很早以前，G.Ewald，D.G.Larman和C.A.Rogers[11]就建立了这样的结论：对于每个给定的C，S(C，X)几乎总是拓扑(k-1)-球面。作为这一结果的后续，彼得·麦克马伦在1974年问道，这些阴影边界中的大多数是否会有finite“Long”[15]。对于多面体，这已经被证明是正确的，当子空间X的维度为1或n-1时，对于一般凸体也是如此。这里，我们证明了几乎所有的阴影边界都具有finite“长度”，而不管子空间X的维度k，0<；k<；n。在finite“长度”的背景下，也考虑了的阴影边界的集合。1989年，P.Gruber和H.Sorger证明了在Baire范畴意义下，大多数(C，X)对(C，X)，其中C是R^n中的凸体，X是R^n的一个(n-1)维子空间，产生finite长度的影子边界S(C，X)。这里我们证明了这一结果对(C，X)也成立，其中X是k维子空间，0<；k<；n。我们还考虑了凸体的1-骨架中增加路的长度。最后，我们以观察到的结果和在fi阴影边界工作中产生的开放问题作为结束。
If C is a convex body in R^n and X is a k-dimensional linear subspace of R^n, we denote by S(C,X) the shadow boundary of C over X which is defined as the collection of all points which belong to C and to one of its tangent (n-k)-flats orthogonal to X. For almost all directions in R^3, the shadow boundary is a curve encompassing the body C. It has been established long ago by G. Ewald, D.G. Larman and C.A. Rogers [11] that, for every given C, S(C,X) is almost always a topological (k-1)-sphere. As a follow on from this result, in 1974 Peter McMullen asked whether most of these shadow boundaries would have finite “length” [15]. This is already shown to be true for polytopes and also true for general convex bodies when the dimension of the subspace X is 1 or n-1. Here we show that almost all shadow boundaries have finite “length” whatever the dimension k, 0< k< n, of the subspace X. The set of shadow boundaries of infinite “length” has also been considered in the context of Baire category. In 1989, P. Gruber and H. Sorger proved that, in the Baire category sense, most pairs (C,X), where C is a convex body in R^n and X an (n-1)-dimensional subspace of R^n, produce shadow boundaries S(C,X) of infinite length. Here we show that this result also holds for pairs (C,X) where X is a k-dimensional subspace, 0< k< n. We also consider the length of increasing paths in the 1-skeleton of a convex body. We conclude with observations and open questions arising from the work on shadow boundaries of the first two chapters.