标题： 关于平面上浮体和浮性体的相似性猜想 显示英文标题

标题： On the homothety conjecture for the body of flotation and the body of buoyancy on a plane

摘要： 我们研究了平面上凸体的几个密切相关的“相似性猜想”。 使用微分几何的现代语言，我们系统地推导了浮体、稳心体和光照体的基本性质。 作为我们结果的直接结论，我们证明了如果浮体与稳心体相似，并且每个浮力弦从边界截取其总仿射弧长的$\frac{1}{3}$，那么$K$是一个椭圆。 我们还提供了H. Auerbach根据苏格兰书中的经典浮体问题得出的古典定理的自然仿射对应物。 特别是，我们得到了J. Bracho、L. Montejano和D. Oliveros引入的Zindler旋转木马的仿射对应物。 显示英文摘要

摘要： We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy, and bodies of illumination. As a direct consequence of our results, we show that if the body of flotation is homothetic to the body of buoyancy, and if every chord of flotation cuts off from the boundary exactly $\frac{1}{3}$ of its total affine arc length, then $K$ is an ellipse. We also provide natural affine counterparts of the classical theorems on the floating body problem from the Scottish Book due to H. Auerbach. In particular, we obtain an affine counterpart of Zindler carousels introduced by J. Bracho, L. Montejano, and D. Oliveros.