Grünbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one. Here, we introduce new measures of symmetry that measure how far a given convex body is from one with “enough symmetries”. To define these new measures of symmetry, we use affine covariant points. We give examples of convex bodies whose affine covariant points a...

The thermodynamic perturbation theory of hard-sphere chains is generalised to derive an equation of state for hard convex body chains. The hard convex body chain equation of state contains two parameters that are related directly and rigorously to the geometry of the hard convex body. The compressibility factors and second virial coefficients of chains composted of prolate spherocylinders, obla...

In this paper, we focus on lower bounds and algorithms for some basic geometric problems in the one-pass (insertion only) streaming model. The problems considered are grouped into three categories — (i) Klee’s measure (ii) Convex body approximation, geometric query, and (iii) Discrepancy Klee’s measure is the problem of finding the area of the union of hyperrectangles. Under convex body approxi...

Let a random simplex in a d-dimensional convex body be the convex hull of d+1 random points from the body. We study the following question: As a function of the convex body, is the expected volume of a random simplex monotone non-decreasing under inclusion? We show that this holds if d is 1 or 2, and does not hold if d ≥ 4. We also prove similar results for higher moments of the volume of a ran...

For a given convex body, find a “small” wall hole through which the convex body can pass. This type of problems goes back to Zindler [14] in 1920, who considered a convex polytope which can pass through a fairly small circular holes. A related topic known as Prince Rupert’s problem can be found in [2]. Here we concentrate on the case when the convex body is a regular tetrahedron or a regular n-...

We prove that no triangular frame can hold a convex body, and a convex body can pass through a triangular hole ∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Applying these result, we prove that a regular tetrahedron of unit edge can pass through an equilateral triangular hole if and only if the edge length of the hole is at least (1 + √ 2)...

We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to innnity. The measure of deviation used is the diierence of the mean width of the convex body and the approximating polytopes. The following results are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for st...

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the NewtonOkounkov convex body o...