论文标题
在凸面上的凸面和同型凸壳功能
On the convex hull and homothetic convex hull functions of a convex body
论文作者
论文摘要
该注释的目的是调查凸壳的特性和浓度$ k $的均匀凸壳功能在欧几里得$ n $空加的范围，定义为$ k $的结合量及其翻译的一个结合，$ k $的体积，以及$ k $的同性恋副本的翻译，分别是$ k $的翻译。特别是，我们证明了身体$ k $的凸船体功能并不确定$ k $。此外，我们证明了Petty提出的极性投影物体问题的等效性，以及G.Horváth和Lángi的猜想，内容涉及凸体的翻译恒定体积特性。我们简短地证明了Jerónimo-Castro的一些定理，介绍了同构凸功能，并证明了翻译恒定体积属性的同一个变体，以$ 3 $二维的凸Polyhedra。我们还应用结果来描述凸体照明体的特性。
The aim of this note is to investigate the properties of the convex hull and the homothetic convex hull functions of a convex body $K$ in Euclidean $n$-space, defined as the volume of the union of $K$ and one of its translates, and the volume of $K$ and a translate of a homothetic copy of $K$, respectively, as functions of the translation vector. In particular, we prove that the convex hull function of the body $K$ does not determine $K$. Furthermore, we prove the equivalence of the polar projection body problem raised by Petty, and a conjecture of G.Horváth and Lángi about translative constant volume property of convex bodies. We give a short proof of some theorems of Jerónimo-Castro about the homothetic convex hull function, and prove a homothetic variant of the translative constant volume property conjecture for $3$-dimensional convex polyhedra. We also apply our results to describe the properties of the illumination bodies of convex bodies.
