Condition number and eccentricity of a closed convex cone

Abstract

We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in R n . The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.

Centers and partial volumes of convex cones II. Advanced topics

“…Non-eccentricity is a geometric property Table 1 Largest observed value (second row) and expected value (third row) of θ(K) n = 3 n = 4 n = 5 n = 10 n = 15 n = 20 0.1988 π 0.2239 π 0.2409 π 0.2479 π 0.2492 π 0.2497 π 0.0528 π 0.0720 π 0.0829 π 0.1025 π 0.1089 π 0.1127 π studied in depth in the references Henrion and Seeger (2011) and Seeger and Torki (2013). The following proposition is a nontrivial result borrowed from (Seeger and Torki 2013, Theorem 2.2).…”

“…According to a terminology used in Henrion and Seeger (2011), a cone K ∈ n is called non-eccentric if ξ(K ) = ξ(K * ). Non-eccentricity is a geometric property Table 1 Largest observed value (second row) and expected value (third row) of θ(K) n = 3 n = 4 n = 5 n = 10 n = 15 n = 20 0.1988 π 0.2239 π 0.2409 π 0.2479 π 0.2492 π 0.2497 π 0.0528 π 0.0720 π 0.0829 π 0.1025 π 0.1089 π 0.1127 π studied in depth in the references Henrion and Seeger (2011) and Seeger and Torki (2013).…”

An illumination problem: optimal apex and optimal orientation for a cone of light

“…A problem closely related to (18) is that of finding a largest ball centered at a unit vector and contained in a given Q ∈ (R n ):…”

“…generated by a finite collection {c i } m i=1 of unit vectors. Suppose that K is pointed and let (y K , s K ) be the solution to (18). Then the boundary E(y K , s K ) = {x ∈ R n : s K x = y K , x } of the revolution cone (y K , s K ) contains at least two of the c i 's.…”

On highly eccentric cones

“…From the general theory of incenters and circumcenters one knows that ξ(K ), η(K ) is positive, where y, x = y T x stands for the usual inner product of R n . Henrion and Seeger (2011) suggest to measure the degree of eccentricity of K in terms of the angle ψ(K ) := arccos ξ (K ), η(K ) between incenter and circumcenter. A large value of ψ(K ) indicates that K is highly eccentric.…”