The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
1answer
192 views

Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets

Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha_2(A)))-\log(\mathrm{Vol}(\alpha_1(A)))$ where $\alpha_1,\alpha_2$ are two affine bijections such ...
5
votes
2answers
426 views

Applications of Cauchy's Arm Lemma

Cauchy's Arm Lemma is used in the proof of Cauchy's Rigidity Theorem for convex polyhedra. The Lemma states that in the plane or on the sphere that if all but one of the side lengths of two convex ...
1
vote
1answer
679 views

Geometric interpretation of singular values

The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and ...
3
votes
1answer
151 views

Edge-maximizing projective transformation on polytopes

Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let projn(e) denote the length of the projection of edge e onto vector n. Consider the set E of ...
11
votes
3answers
613 views

Continuous automorphism groups of normed vector spaces?

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
5
votes
2answers
668 views

Maximal Ellipsoid

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ...
1
vote
1answer
371 views

Is the direction of the longest line of a polytope unique?

The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. The affine subspace is given by: $X \mbox{ u} = y$ where $u$ ...
5
votes
0answers
407 views

Generalizations of generators / hyperplanes descriptions for cones to partially-ordered fields?

Background: given a finite-dimensional real vector space V of dimension d, I can define a pointed cone in two ways: either as a set of the form $\{r_1v_1 + \cdots + r_nv_n \mid r_1, \dots, r_n \in ...
14
votes
9answers
6k views

Algorithm for finding the volume of a convex polytope

It's easy to find the area of a convex polygon by division into triangles, but what is the optimal way of finding the volume of higher-dimensional convex bodies? I tried a few methods for dividing ...