A sum of squares theorem for visibility

We extend John's inscribed ellipsoid theorem, as well as Loewner's circumscribed ellipsoid theorem, from convex bodies to proper cones. To be more precise, we prove that a proper cone $$K$$K in $$\mathbb {R}^n$$Rn contains a unique ellipsoidal cone $$Q^\...

Let k 4 . A finite planar point set X is called a convex k -clustering if it is a disjoint union of k sets X ₁ , . . . ,X _k of equal sizes such that x ₁ x ₂ . . . x _k is a convex k -gon for each choice of x ₁ ...

The Colorful Carathéodory theorem by Bárány (1982) states that given d + 1 sets of points in R d , the convex hull of each containing the origin, there exists a simplex (called a 'rainbow simplex') with at most one point from each point set, which also ...