Introduction 3 1. Affine geometry 4 1.1. Affine spaces 5 1.1.1. Euclidean geometry and its isometries 5 1.1.2. Affine spaces 7 1.1.3. Affine transformations 8 1.1.4. Tangent spaces 9 1.1.5. Acceleration and geodesics 10 1.1.6. Connections 11 1.2. The hierarchy of structures 11 1.3. Affine vector fields 12 1.4. Affine subspaces 13 1.5. Volume in affine geometry 14 1.6. Centers of gravity 14 1.7....

Let ∆m = {(t0, . . . , tm) ∈ R : ti ≥ 0, ∑m i=0 ti = 1} be the standard m-dimensional simplex. Let ∅ 6= S ⊂ ⋃ ∞ m=1 ∆m, then a function h : C → R with domain a convex set in a real vector space is S-almost convex iff for all (t0, . . . , tm) ∈ S and x0, . . . , xm ∈ C the inequality h(t0x0 + · · ·+ tmxm) ≤ 1 + t0h(x0) + · · ·+ tmh(xm) holds. A detailed study of the properties of S-almost convex...

Let ∆m = {(t0, . . . , tm) ∈ Rm+1 : ti ≥ 0, ∑m i=0 ti = 1} be the standard m-dimensional simplex and let ∅ = S ⊂ ⋃∞ m=1 ∆m. Then a function h : C → R with domain a convex set in a real vector space is S-almost convex iff for all (t0, . . . , tm) ∈ S and x0, . . . , xm ∈ C the inequality h(t0x0 + · · ·+ tmxm) ≤ 1 + t0h(x0) + · · ·+ tmh(xm) holds. A detailed study of the properties of S-almost co...

A graph has strong convex dimension 2 if it admits a straightline drawing in the plane such that its vertices form a convex set and the midpoints of its edges also constitute a convex set. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension 2 are planar and therefore have at most 3n− 6 edges. We prove that all such graphs have indeed at most 2n − 3 edges, while on the o...

polytope Abstract polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytope November, 2013 – p. 2 Abstract polytope Abstract polytope −→ combinatorial generalization of convex polytopepolytope Abstract polytope −→ combinatorial generalization of convex polytop...

ALGEBRAIC GEOMETRY 79 It may be advisable to give a special name to those varieties which admit every (allowable) ground field as field of definition. Obviously, these are the varieties which are defined over the prime field of the given characteristic p. I propose to call them universal varieties. The projective space and the Grassmannian varieties are examples of universal varieties. Another ...

Let K ⊂ R d be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull I L (K) = conv(K ∩L), where L is a randomly translated and rotated copy of the integer lattice Z d. We estimate the expected number of vertices of I L (K), whose behaviour is similar...

A finite set of points in the plane is described as in convex position if it forms the set of vertices of a convex polygon. This work studies the ratio between the maximum area of convex heptagons with vertices in P and the area of the convex hull of P, where the planar point set P is in convex position.

Real Algebraic Geometry in Convex Optimization

We consider all planar oriented curves that have the following property depending on a xed angle '. For each point B on the curve, the rest of the curve lies inside a wedge of angle ' with apex in B. This property restrains the curve's meandering, and for ' 2 this means that a point running along the curve always gets closer to all points on the remaining part. For all ' < , we provide an upper...