照亮凸体边界的平行光束的一致压缩

Uniform Compressing of Parallel Light Beams Illuminating the Boundary of Convex Bodies

鉴于凸体边界的平行光照亮问题与关于凸体覆盖的Hadwiger猜想之间的密切联系,证明了照亮凸体边界的平行光束的宽度可以被一致地压缩,并借此在不使用归纳法的前提下证明了覆盖凸体所需小位似体的最小数目等于照亮凸体边界的平行光源的最小个数.

In view of the close relation between illuminating boundaries of convex bodies and Hadwiger＇s covering conjecture, it is proved that parallel light beams illuminating the boundary of a convex body can be uniformly compressed. Based on this, a new proof, without using induction, is given to show that the least number of smaller homothetic copies of a convex body needed to cover this convex body is equal to the least number of parallel light sources needed to illuminate its boundary.