Ulam floating functions

Ulam Floating Body

We study a new construction of bodies from a given convex body in $\mathbb{R}^{n}$ which are isomorphic to (weighted) floating bodies. We establish several properties of this new construction, including its relation to $p$-affine surface areas. We show that these bodies are related to Ulam's long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.

中心对称下平面Ulam浮体问题中周长密度σ = 1/8、3/8时的刚性

Rigidity in the planar Ulam floating body problem with perimetral densities $σ=\tfrac18,\tfrac38$ under central symmetry

研究平面Ulam浮体问题，针对周长密度σ = 1/8或3/8，通过证明得出中心对称且具C^1边界的严格凸平衡漂浮体K唯一为圆盘。

平面乌拉姆浮体问题中周长密度σ=1/6的刚性

Rigidity in the Planar Ulam Floating Body Problem with perimetral density $σ=\tfrac16$

研究平面凸域在周长密度σ=1/6下的乌拉姆浮体问题，利用Zindler旋转装置框架，将问题转化为与内接正六边形相关的二维动力系统。主要结果表明，圆盘是唯一在该周长密度下在任意位置都能保持平衡的凸域，为有理周长密度的凸设定提供了新的刚性结果。

On bodies floating in equilibrium in every orientation

Ulam's problem 19 from the Scottish Book asks: {\it is a solid of uniform density which floats in water in every position necessarily a sphere?} We obtain several results related to this problem.

Convex Floating Bodies of Equilibrium

We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.

Convex bodies generated by sublinear expectations of random vectors

We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined convex bodies.