# convex set proof example

[(Theorem)-375.9(3.2.5)]TJ Notice that while deﬁning a convex set, (S)Tj /F2 1 Tf (S)Tj /F2 1 Tf /F2 1 Tf /F2 1 Tf /F3 1 Tf S )Tj /F2 1 Tf 0.2731 Tc 0 -2.3625 TD (=)Tj 1.7506 0 TD 0.0001 Tc (. 14.269 0 TD >> 2.8875 0 TD /F2 1 Tf (b)Tj /F2 1 Tf Closed convex sets are convex sets that contain all their limit points. A vector x0 is an interior point of the set X, if there is a ball B(x0,r) contained entirely in the set X Def. /F2 1 Tf 0.2779 Tc /F3 1 Tf ()Tj /F2 1 Tf -2.3744 -5.9277 TD [(of)-301.8(the)-301.9(s)0(mallest)-301.9(ane)-301.9(subset)]TJ Linear algebra proof that this set is convex mathematics stack. [(Car)50.1(a)-0.1(th)24.8(´)]TJ [(de“nitions)-301.8(ab)-26.1(out)-301.8(cones. 3.8079 0 TD 0.0001 Tc 0.5893 0 TD (,)Tj )-558.9(T)0.1(he)-386.6(family)]TJ ()Tj 0.3338 0 TD 0.6669 0 TD 0 Tc S 0 -1.2052 TD 0 Tc (E)Tj /F2 1 Tf 14.3552 0 TD 0 G (S)Tj 4.4443 0 TD 9.3037 0 TD 0 Tw /F4 1 Tf 0 Tc /F7 10 0 R 0.5101 0 TD 0 -1.2057 TD (Š)Tj 1.386 0 TD /F5 1 Tf /F2 1 Tf /F5 1 Tf 0.0001 Tc 14.3462 0 0 14.3462 187.893 330.0511 Tm /F2 5 0 R 0.6608 0 TD 0.7379 0 TD 1.0554 0 TD (E)Tj /F2 1 Tf -20.8312 -1.2052 TD 20.6626 0 0 20.6626 237.609 626.313 Tm 5.5102 0 TD 15.1802 0 TD 14.3462 0 0 14.3462 458.802 515.6041 Tm [(Car)50.1(a)-0.1(th)24.8(´)]TJ 0 Tc (b)Tj /F4 1 Tf /F2 1 Tf 1.494 w (i)Tj 20.6626 0 0 20.6626 119.43 468.894 Tm [(is)-306.8(a)-307(c)50.2(onvex)-306.9(c)50.2(o)0(mbina-)]TJ ()Tj ⁡ 0 Tc ()Tj -9.8325 -1.2052 TD /F4 1 Tf (. [(,)-349.8(and)]TJ 0.6608 0 TD More explicitly, a convex problem is of the form min f (x) s.t. /F4 1 Tf ⁡ The definition of a convex set and a convex hull extends naturally to geometries which are not Euclidean by defining a geodesically convex set to be one that contains the geodesics joining any two points in the set. /F5 1 Tf [(are)-301.9(t)0(he)-301.9(\(closed\))-301.9(half)-301.8(s)0(paces)-301.9(asso)-26.2(ciated)-301.9(with)]TJ [(,)-360.7(for)-358.4(any)-358.2(\(nonempty\))-357.8(family)]TJ The elements of are called convex sets and the pair (X, ) is called a convexity space. /ExtGState << -1.7998 -1.2057 TD -7.9956 -2.363 TD 0 Tc [(\))-350(i)0(s)-350(t)0.2(he)-349.6(c)50.2(onvex)-350.1(hul)-50(l)-350.1(of)]TJ 0.8886 0 TD /F8 16 0 R They can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane). 0 -1.2057 TD (S)Tj /F2 1 Tf ()Tj [(There)-254.8(is)-254.8(also)-254.9(a)-254.9(v)26.1(ersion)-254.5(o)-0.1(f)-255.2(T)-0.2(heorem)-254.6(3.2.2)-254.9(f)0(or)-254.8(con)26(v)26.1(ex)-254.4(cones. /F4 1 Tf Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). Convex sets and convex functions. >> /F5 1 Tf 0.3541 0 TD 1.782 0 TD 0.3541 0 TD [(with)-301.8(1)]TJ [(The)-263(f)0.1(ollo)26.2(wing)-263(tec)26.2(hnical)-262.9(\()0.1(and)-263.1(dull!\))-393.2(lemma)-263(pla)26.2(y)0(s)-263(a)-263(crucial)]TJ (|)Tj /F1 1 Tf /F6 9 0 R 0 Tc (\()Tj ET 0.2496 0 TD 0 g ()Tj 0.2779 Tc /GS1 gs 0 g /F5 1 Tf 0.3541 0 TD [(L,)-333.7(I)]TJ If a and b are points in a vector space the points on the straight line between a and … [(c)50.1(onvex)]TJ BT 20.6626 0 0 20.6626 72 702.183 Tm 0.6608 0 TD (\()Tj /F4 1 Tf 0.6608 0 TD 1.369 0 TD >> /F2 1 Tf 0.7836 0 TD ET 20.6626 0 0 20.6626 208.116 406.2631 Tm /F4 1 Tf /F4 1 Tf 34 0 obj (i)Tj 5.2758 0 TD /F1 1 Tf 0 Tw )-762.5(CONVEX)-326(SETS)]TJ (f)Tj (i)Tj [(union)-375.5(of)-375.4(triangles)-375.5(\(including)-375.5(in)26(terior)-375.5(p)-26.2(oin)26(ts\))-375.5(whose)-375.5(v)26.1(er-)]TJ 0 Tc -20.5425 -2.941 TD 0.2779 0 TD 8.3171 0 TD /F8 16 0 R 0 Tc ()Tj 2.1087 0 TD /F5 1 Tf (E)Tj /F4 1 Tf 0 Tc (f)Tj /F2 1 Tf 20.6626 0 0 20.6626 124.938 436.3051 Tm (f)Tj The notion of convexity may be generalised to other objects, if certain properties of convexity are selected as axioms. (de“ning)Tj [(tices)-301.9(b)-26.2(elong)-301.9(to)]TJ [5][6], The complement of a convex set, such as the epigraph of a concave function, is sometimes called a reverse convex set, especially in the context of mathematical optimization. /F5 1 Tf /F4 1 Tf (I,)Tj (i)Tj 0 Tc 2.0442 0 TD (a)Tj /F2 1 Tf /Length 2864 (i)Tj [(Theorem)-375.9(3.2.2)]TJ 0.967 0 TD 8 B [(\)\()446(o)445.9(r)]TJ 0.1237 -0.7932 TD 0.5101 0 TD (Š)Tj [(Observ)26.2(e)-398.9(t)0.1(hat)-398.9(if)]TJ /F2 1 Tf [(F)78.6(o)0(r)-327.5(t)0.1(his)-327.5(reason,)-333.9(w)26.1(e)-327(will)-327.4(also)-327.5(sa)26.2(y)-327.5(t)0.1(hat)]TJ (�s. 0.3615 Tc /F2 1 Tf << 0 Tc /F2 1 Tf 5.3451 0 TD 391.038 676.846 l 3.4799 0 TD 387.355 636.114 l (is)Tj 0.5549 0 TD (+)Tj Proof: This is straightforward from the de nition. 2.7455 0 TD 0.6991 0 TD 220.959 591.807 l )]TJ ()Tj 9.9092 0 TD 0 Tc ET stream ()Tj /F4 1 Tf (v)Tj (S)Tj (? 0.0001 Tc 1.4008 0 TD 0.876 0 TD Many algorithms for convex optimization iteratively minimize the function over lines. 0.3999 0 TD /F6 1 Tf /F2 1 Tf 0.2777 Tc 0.0001 Tc [(Bounded)-263.2(c)0(on)26(v)26.1(e)0(x)-263.2(sets)-263.5(arising)-263.6(a)-0.1(s)-263.1(t)0(he)-263.6(in)26(tersection)-263.2(o)-0.1(f)-263.5(a)-263.6(“nite)]TJ 0.5711 0 TD >> >> (\))Tj /F3 1 Tf (a)Tj )Tj x. in. 20.6626 0 0 20.6626 453.762 626.313 Tm /F4 1 Tf 0 Tc -0.0001 Tc /GS1 gs 20.6626 0 0 20.6626 347.589 529.6981 Tm [(is)-267.9(a)-268.4(“)0.1(nite)-267.9(\(of)-267.8(i)0(n“nite\))-268.3(set)-267.9(of)-267.8(p)-26.2(o)-0.1(in)26(ts)-268.3(in)-268(the)-267.9(a)-0.1(ne)-267.9(p)-0.1(lane)]TJ /F2 1 Tf 379.786 629.139 m /F8 1 Tf [(\). (\))Tj [(3.2. 226.093 685.464 200.694 710.863 169.4 710.863 c 1.2715 0 TD (i)Tj Such an affine combination is called a convex combination of u1, ..., ur. 0.3338 0 TD /Font << [14][15], The Minkowski sum of two compact convex sets is compact. S 0.9857 0 TD 0.3809 0 TD 0.3549 Tc ()Tj 1.2113 0.95 TD (b)Tj /F2 1 Tf 0 g 14.3462 0 0 14.3462 216.234 261.6151 Tm /F2 1 Tf 0 Tc (H)Tj %âãÏÓ /F4 1 Tf 0 Tc 20.6626 0 0 20.6626 149.112 626.313 Tm 0 Tc 0 Tc /F2 1 Tf 14.3462 0 0 14.3462 187.416 587.3701 Tm /F4 1 Tf [(,i)366.7(f)]TJ -1.4409 3.3061 TD 0 0 1 rg /F3 1 Tf 0 1 0 rg In a real vector-space, the Minkowski sum of two (non-empty) sets, S1 and S2, is defined to be the set S1 + S2 formed by the addition of vectors element-wise from the summand-sets, More generally, the Minkowski sum of a finite family of (non-empty) sets Sn is the set formed by element-wise addition of vectors, For Minkowski addition, the zero set {0} containing only the zero vector 0 has special importance: For every non-empty subset S of a vector space, in algebraic terminology, {0} is the identity element of Minkowski addition (on the collection of non-empty sets).[13]. s 14.3462 0 0 14.3462 338.004 254.973 Tm ()Tj (\). [(of)-359.4(dimen-)]TJ (i)Tj Concretely the solution set to (4.6) is cone. [(\),)-236(and)-219.2(similarly)-219.6(for)]TJ /F5 1 Tf 20.6626 0 0 20.6626 72 702.183 Tm (]=)Tj 0.8912 0 TD (? 12.9565 0 TD ()Tj 0.1666 Tc [(\)i)283.7(st)283.6(h)283.5(e)]TJ 0.4587 0 TD (\()Tj /F2 1 Tf 0 Tc [(sp)-26.2(ecial)-301.8(p)-0.1(rop)-26.2(erties? /Font << ({)Tj 4.2217 0 TD endobj 0 Tc 0.4587 0 TD endobj /F4 1 Tf 0.4504 Tc 387.657 628.847 l 20.6626 0 0 20.6626 492.642 375.2401 Tm 1.0084 0 TD 0.7836 0 TD 1.3699 0 TD 20.6626 0 0 20.6626 72 701.031 Tm /F4 1 Tf 0.4617 0 TD 0.0001 Tc ()Tj 0.9722 -1.7101 TD (S)Tj /F4 1 Tf /GS1 11 0 R /GS1 11 0 R 5.5685 0 TD /F2 1 Tf 0.3541 0 TD [(ev)26.1(ery)-298.9(p)-26.2(o)-0.1(in)26(t)]TJ /F4 1 Tf 329.211 654.17 l (m)Tj /F3 1 Tf /F2 1 Tf (1)Tj /F5 1 Tf 20.6626 0 0 20.6626 379.566 407.8741 Tm /F4 1 Tf 11.9551 0 0 11.9551 289.53 684.819 Tm