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Register a new userUnique local determination of convex bodies
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Barker and Larman asked the following. Let $K subset {Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K subset {Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K cap H$ be given. Is $K$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the caps cut off from $K$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C^2_+$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1 le k le d-1$ or for $1 le k le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1 le k le l le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K$.
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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.
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial complexity of the bodies and the topological complexity of their realization space. First, we show that every combinatorial type is realizable and its realization space is contractible under mild assumptions. Second, we prove a universality theorem that says the restriction of the realization space to arrangements polygons with a bounded number of vertices can have the homotopy type of any primary semialgebraic set.
Let $K$ and $L$ be two convex bodies in $mathbb R^n$, $ngeq 2$, with $Lsubset text{int}, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $lambda$. J. Barker and D. Larman proved that if $L$ is a ball, then $K$ is a ball concentric with $L$. In this paper we prove that there exist an infinite number of closed curves, different from circles, which possess an equichordal convex body. If the dimension of the space is more than or equal to 3, then only Euclidean balls possess an equichordal convex body. We also prove some results about isoptic curves and give relations between isoptic curves and convex rotors in the plane.
Let $K$ be an isotropic symmetric convex body in ${mathbb R}^n$. We show that a subspace $Fin G_{n,n-k}$ of codimension $k=gamma n$, where $gammain (1/sqrt{n},1)$, satisfies $$Kcap Fsubseteq frac{c}{gamma }sqrt{n}L_K (B_2^ncap F)$$ with probability greater than $1-exp (-sqrt{n})$. Using a different method we study the same question for the $L_q$-centroid bodies $Z_q(mu )$ of an isotropic log-concave probability measure $mu $ on ${mathbb R}^n$. For every $1leq qleq n$ and $gammain (0,1)$ we show that a random subspace $Fin G_{n,(1-gamma )n}$ satisfies $Z_q(mu )cap Fsubseteq c_2(gamma )sqrt{q},B_2^ncap F$. We also give bounds on the diameter of random projections of $Z_q(mu )$ and using them we deduce that if $K$ is an isotropic convex body in ${mathbb R}^n$ then for a random subspace $F$ of dimension $(log n)^4$ one has that all directions in $F$ are sub-Gaussian with constant $O(log^2n)$.
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