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Convex Geometry

Csaba Vincze (2013)

University of Debrecen

7.4 Excercises

7.4 Excercises

Excercise 7.4.1 Generalize the nearest-point-type argumentation by substituting p with a closed convex subset in the coordinate space of dimension n.

Excercise 7.4.2 Prove that if K is a closed convex set then for any point q in the space there exists a uniquely determined point in K which is the nearest point of K to q. In other words closed convex sets satisfy the nearest-point property with respect to the Euclidean distance.

Excercise 7.4.3 Find a convex set such that it does not satisfy the nearest-point property with respect to the taxicab norm.

Excercise 7.4.4 Find the extreme points of the convex hull of the points

( 0,0 ) , ( 0,1 ) , ( 1,2 ) , ( 2,3 ) , ( 3,3 ) , ( 3,0 ) , ( 2,1 )

in the coordinate plane.

Excercise 7.4.5 Prove that for any compact convex set in the coordinate plane the profile is a closed set.

Excercise 7.4.6 Find an example for a compact convex set in the coordinate space of dimension three such that the profile is not closed.

Hint. Divide a cylinder into two parts by a plane containing the axis.

Excercise 7.4.7 Prove that the maximum of a continuous convex function on a compact convex set is attained at one of its extreme points.

Hint. Suppose, in contrary, that the statement is false and represent the elements as convex combinations of some extreme points. The convexity of the function gives an upper bound for the values of the function in terms of the values at the extreme points.

Excercise 7.4.8 Find the set of extreme points of a closed half-space.

Excercise 7.4.9 Let K be a closed convex subset in the coordinate space of dimension n and suppose that the hyperplane H supports K at the point p. Prove that p is an extreme point of the intersection of K and H if and only if it is an extreme point of K.

Excercise 7.4.10 Prove that the Minkowski functional is positively 1-homogeneous and subadditive: for any positive real number t

l ( t v ) = t l ( v )   a n d   l ( v + w ) l ( v ) + l ( w ) .

Excercise 7.4.11 Prove that each norm is just the Minkowski functional induced by the unit ball.

Excercise 7.4.12 Find the Minkowski functional l induced by the convex hull of the points

( 1,0 ) , ( 0,1 ) , ( - 1,0 ) , ( 0 , - 1 ) ;

see the taxicab norm 1.48.

Excercise 7.4.13 Find the Minkowski functional l induced by the convex hull of the points

( 1,1 ) , ( - 1,1 ) , ( - 1 , - 1 ) , ( 1 , - 1 ) ;

see the maximum norm.

Excercise 7.4.14 Find the Minkowski functional induced by an ellipse centered at the origin. How to deduce l as a norm coming from an inner product?

Excercise 7.4.15 Find a symmetric convex body K such that the induced Minkowski space has the minimal set of isometries.

Excercise 7.4.16 Find the Minkowski functional associated to the closed disk

( x - 1 ) 2 + ( y - 1 ) 2 4

and prove that the reflection about the line joining the origin and the center of the disk is a linear isometry of the Minkowski space.

Excercise 7.4.17 Prove that the domain of the support function is convex.

Excercise 7.4.18 Prove that the support function is a positively 1-homogeneous convex function on its domain.

Excercise 7.4.19 Find the polar set of the convex hull of the points

( 1,2 ) , ( 2 , - 1 ) , ( - 3 , - 1 )

in the coordinate plane.

Excercise 7.4.20 Find the polar set of the convex hull of the points

( 1,1 ) , ( - 1,1 ) , ( - 1 , - 1 ) , ( 1 , - 1 )

in the coordinate plane.

Excercise 7.4.21 Find the polar sets of disks and ellipses centered at the origin in the coordinate plane.

Excercise 7.4.22 Find the polar set of a tetrahedron in the coordinate space of dimension three.

Excercise 7.4.23 Find the polar set of the cube with vertices

( 1,1 , 1 ) , ( 1,1 , - 1 ) , ( - 1,1 , 1 ) , ( - 1,1 , - 1 ) , ( - 1 , - 1,1 ) ,

( - 1 , - 1 , - 1 ) , ( 1 , - 1,1 ) , ( 1 , - 1 , - 1 )

in the coordinate space of dimension three.

Excercise 7.4.24 Find the polar set of the octahedron with vertices

( 1,0 , 0 ) , ( 0,1 , 0 ) , ( - 1,0 , 0 ) , ( 0 , - 1,0 ) , ( 0,0 , 1 ) , ( 0,0 , - 1 )

in the coordinate space of dimension three.

Excercise 7.4.25 Prove that the polar set is a closed convex set containing the origin.

Hint. For the basic properties of polar sets see chapter 9.

Excercise 7.4.26 Find the support function of the convex hull of the points

( 1,2 ) , ( 2 , - 1 ) , ( - 3 , - 1 )

in the coordinate plane.

Excercise 7.4.27 Find the support function of the convex hull of the points

( 1,1 ) , ( - 1,1 ) , ( - 1 , - 1 ) , ( 1 , - 1 )

in the coordinate plane.

Excercise 7.4.28 Find the support functions of disks and ellipses centered at the origin in the coordinate plane.

Excercise 7.4.29 Find an example to illustrate that the normality in Minkowski planes is not a symmetric relation in general.

Hint. Consider squares (see the taxicab norm or the maximum norm) or regular octagons in the plane.

Excercise 7.4.30 Prove that ellipses and regular hexagons areRadon curves.

Excercise 7.4.31 Prove that the polar of a Radon curve is also a Radon curve.

A differentiable curve

c : [ a , b ] E 2

is parameterized by Minkowskian arclength if its derivatives have unit length with respect to the Minkowski functional at each parameter s, i.e. l(c'(s))=1. Let c be a twice-differentiable parametrization of the unit circle of a Minkowski plane by Minkowski arclength.

Excercise 7.4.32 Prove that c is a Radon curve if and only if the Wronskian

s d e t ( c ( s ) , c ' ( s ) )

is constant.

Hint. Using the Minkowskian arclenght parametrization c'(s)=c(t(s)). The Radon curve property requires that c'(t(s)) is parallel to c(s). By differentiation we have that c''(s) is parallel to c(s) which is equivalent to the vanishing of the derivative of the Wronskian determinant.

Excercise 7.4.33 Prove that for any Radon-curve c the line joining the origin and c(s) sweeps out equal areas during equal intervals of time.

Hint. Use that

A ( t ) = 1 2 a t s d e t ( c ( s ) , c ' ( s ) ) d s

gives the area swept out from a to t; see Kepler's second law of planetary motions.