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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 Ulams long-standing floating body problem which asks whether Euclidean balls are the only bodies that can float, without turning, in any orientation.
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We investigate weighted floating bodies of polytopes. We show that the weighted volume depends on the complete flags of the polytope. This connection is obtained by introducing flag simplices, which translate between the metric and combinatorial structure. Our results are applied in spherical and hyperbolic space. This leads to new asymptotic results for polytopes in these spaces. We also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in Euclidean space.
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.
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Feffermans invariant hypersurface measure.
Given any two convex polyhedra P and Q, we prove as one of our main results that the surface of P can be reshaped to a homothet of Q by a finite sequence of tailoring steps. Each tailoring excises a digon surrounding a single vertex and sutures the digon closed. One phrasing of this result is that, if Q can be sculpted from P by a series of slices with planes, then Q can be tailored from P. And there is a sense in which tailoring is finer than sculpting in that P may be tailored to polyhedra that are not achievable by sculpting P. It is an easy corollary that, if S is the surface of any convex body, then any convex polyhedron P may be tailored to approximate a homothet of S as closely as desired. So P can be whittled to e.g., a sphere S. Another main result achieves the same reshaping, but by excising more complicated shapes we call crests, still each enclosing one vertex. Reversing either digon-tailoring or crest-tailoring leads to proofs that any Q inside P can be enlarged to P by cutting Q and inserting and sealing surface patches. One surprising corollary of these results is that, for Q a subset of P, we can cut-up Q into pieces and paste them non-overlapping onto an isometric subset of P. This can be viewed as a form of unfolding Q onto P. All our proofs are constructive, and lead to polynomial-time algorithms.
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