(S)Tj >> (\()Tj [(W)78.7(e)-377.6(shall)-377.1(p)0(ro)26.2(v)26.2(e)-377.6(that)]TJ 0 Tc 14.3462 0 0 14.3462 325.017 573.402 Tm /F2 1 Tf 0.876 0 TD (c)Tj /F4 1 Tf /F5 1 Tf [(,)-349.8(and)]TJ /F4 1 Tf /F7 1 Tf /F4 1 Tf [(,)-315.4(t)0.2(hat)-306.9(is,)]TJ -19.4754 -1.2057 TD 0 -1.2052 TD (K)Tj [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ (S)Tj /F7 1 Tf 9.1665 0 TD 1.0554 0 TD 3 /F7 10 0 R ( [(\),)-236(and)-219.2(similarly)-219.6(for)]TJ << 20.6626 0 0 20.6626 195.444 292.4041 Tm 0 Tc (H)Tj /F4 1 Tf (v)Tj /GS1 gs ()Tj (i)Tj /F2 1 Tf s 0.514 0 TD BT [(3.2. /F5 1 Tf 4.7928 0 TD [(\(namely)78.4(,)-393.2(t)0.1(he)]TJ (a)Tj 0.0001 Tc 0 Tc 0 Tc Lecture 3: september 4 3. [(CHAPTER)-327.3(3. How can we make them convex? 0.0001 Tc 0.0001 Tc 0.1666 Tc endstream [(p)-26.2(o)-0.1(in)26(ts)-301.9(in)26(v)26.1(o)-0.1(lv)26.1(ed)-301.9(in)-301.9(the)-301.9(c)0(on)26(v)26.1(e)0(x)-301.5(c)0(om)25.9(binations? [(the)-301.9(f)0(ollo)26.1(wing)-301.9(result:)]TJ )Tj [(. ()Tj (f)Tj (E)Tj 0 Tc ()Tj /F4 1 Tf [(,)-558(o)0(f)-516.6(di-)]TJ 0 Tc /F4 1 Tf /F4 1 Tf >> 20.3985 0 TD (\()Tj The subspace Y is a convex set if for each pair of points a, b in Y such that a ≤ b, the interval [a, b] = {x ∈ X | a ≤ x ≤ b} is contained in Y. 0 Tc 0.2781 Tc 0 0 1 rg (I)Tj /F3 1 Tf 357.557 625.823 l -0.0001 Tc (i)Tj (a)Tj (f)Tj /F2 1 Tf 0.6904 0 TD /F3 1 Tf 33 0 obj /F2 1 Tf 0.5798 0 TD /F2 1 Tf (in)Tj 9.3037 0 TD 0.5893 0 TD 0.4586 1.7101 TD -19.3423 -1.2052 TD (I)Tj 14.3462 0 0 14.3462 339.822 487.911 Tm /F4 1 Tf /F3 1 Tf (. /F5 1 Tf /F4 1 Tf (=0)Tj -15.875 -1.2052 TD /F3 1 Tf We say a set Cis convex if for any two points x;y2C, the line segment (1 )x+ y; 2[0;1]; lies in C. The emptyset is also regarded as convex. -18.3735 -2.363 TD 0.3541 0 TD /F5 1 Tf 0 g 11.8754 0 TD [(has)-393.7(dimension)]TJ 1.001 0 TD (\()Tj /F4 1 Tf 0 g [(hul)-50.1(l)]TJ (i)Tj 5.139 0 TD /F7 1 Tf -4.2496 -1.2052 TD /F3 1 Tf 0.7836 0 TD /F4 1 Tf [(Pr)50.1(o)50(o)-0.1(f)-350.3(s)0.1(ketch)]TJ (})Tj -22.3781 -1.7837 TD /F4 1 Tf endobj (S)Tj (I)Tj >> ({)Tj [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ 1.0554 0 TD The elements of are called convex sets and the pair (X, ) is called a convexity space. /F3 1 Tf /F4 1 Tf /F8 1 Tf >> [(CHAPTER)-327.3(3. /F5 1 Tf 14.3462 0 0 14.3462 125.127 490.701 Tm 20.6626 0 0 20.6626 443.367 529.6981 Tm /F4 1 Tf 0 -1.2052 TD 0.3541 0 TD /F2 1 Tf ({)Tj (K)Tj /F4 1 Tf 5.9912 0 TD [(pro)26.1(v)26.1(e)-359.8(it)-360.2(here. /F2 1 Tf (i)Tj [(can)-377.2(b)-26.1(e)-377.6(w)-0.1(ritten)-377.2(as)-377.1(a)-377.2(c)0.1(on)26.1(v)26.2(e)0.1(x)-377.2(c)0.1(om-)]TJ [(tices)-301.9(b)-26.2(elong)-301.9(to)]TJ 0 g /F9 1 Tf 0 Tc /F2 1 Tf {\displaystyle \operatorname {rec} S} /F4 1 Tf [(in“nite\))-301.9(of)-301.8(con)26(v)26.1(ex)-301.9(sets)-301.9(is)-301.9(con)26(v)26.1(ex. (H)Tj f 0 Tc /GS1 gs [(The)-437.3(answ)26(e)0(r)-436.9(i)0(s)-437.8(y)26.1(es)-437.3(in)-437.4(b)-26.2(o)-0.1(th)-437(cases. 1.2216 0.7187 TD -20.5425 -2.941 TD 0.0001 Tc (i)Tj /F9 1 Tf /F4 1 Tf 17.5298 0 TD [(has)-330.5(“nite)-330.5(supp)-26.1(ort)-330.1(\()0.1(all)]TJ 0 Convex Sets. 0.3615 Tc [(W)78.6(e)-290.6(get)-290.5(t)0(he)-290.1(feeling)-290.6(t)0(hat)-290.5(triangulations)-290.1(pla)26.1(y)-290.6(a)-290.1(crucial)-290.5(r)0(ole,)]TJ 6.5129 0 TD (and)Tj /F5 1 Tf 379.485 628.847 m /F3 1 Tf << BT stream = (Š)Tj 0 Tc (|)Tj /F7 1 Tf 0 Tc 0 Tc 0 Tc 112.707 597.477 m /F5 1 Tf -0.1302 -0.2529 TD 0.0001 Tc /F2 1 Tf (sion)Tj 1.7998 0 TD endstream /F4 1 Tf (Š)Tj 0.0001 Tc /F4 1 Tf /F2 1 Tf A subset C of S is convex if, for all x and y in C, the line segment connecting x and y is included in C. This means that the affine combination (1 − t)x + ty belongs to C, for all x and y in C, and t in the interval [0, 1]. 4.8219 0 TD /F5 1 Tf 14.3462 0 0 14.3462 448.479 623.217 Tm 0 -1.2052 TD /F4 1 Tf 0.0001 Tc 0.8564 0 TD 11.9551 0 0 11.9551 306.315 684.819 Tm The notion of a convex set can be generalized as described below. 0 Tc [(tion)-349.8(of)-349.8(the)]TJ 1 i (+1)Tj 20.6626 0 0 20.6626 533.799 199.0651 Tm 3.6454 0 TD /F4 1 Tf /F4 1 Tf 0.8359 0 TD 1.1665 0 TD (E)Tj (\()Tj [(,)-427.2(t)0(here)-402.1(is)-402.5(a)]TJ /F2 1 Tf << 6.4362 0 TD 20.6626 0 0 20.6626 72 702.183 Tm (=\()Tj (f)Tj /F2 1 Tf 9.1752 0 TD ()Tj (q)Tj -14.333 -1.2052 TD 20.6626 0 0 20.6626 72 702.183 Tm /F2 1 Tf (0)Tj Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field. /F7 1 Tf /F2 1 Tf 20.6626 0 0 20.6626 72 517.845 Tm 1.1255 0 TD 2.3979 0 TD /F9 1 Tf 14.3462 0 0 14.3462 155.538 573.402 Tm 0.9975 0 TD 0.7235 0 TD [(any)-349.9(family)]TJ 1.1168 0 TD (sets,)Tj 0 Tc (and)Tj 0.0001 Tc (\). (m)Tj 0.6943 0 TD 14.3462 0 0 14.3462 161.964 548.499 Tm /F4 1 Tf /Font << /GS1 gs Convex set. 14.3462 0 0 14.3462 458.802 515.6041 Tm /Length 5100 (j)Tj 1.1604 0 TD >> ()Tj ()Tj (E)Tj [(to)-452.2(the)-452.1(s)0.1(et)-452.1(of)-452.2(p)50(o)-0.1(sitive)-452.1(c)50.1(o)-0.1(mbinations)-451.6(o)-0.1(f)-451.8(families)-451.6(o)-0.1(f)]TJ 1.1194 0 TD (. (i)Tj (b)Tj (H)Tj 15 0 obj 1.8726 0 TD (. [(,o)273(r)]TJ (S)Tj 0.0001 Tc 4.8503 0 TD 0.0229 Tc /F4 1 Tf /F2 1 Tf [7], Given r points u1, ..., ur in a convex set S, and r 1.782 0 TD 0.0001 Tc 20.6626 0 0 20.6626 72 701.0491 Tm (C)Tj 0.0001 Tc ()Tj /F4 1 Tf 0.1667 Tc /F5 1 Tf 3.0212 0 TD /F3 1 Tf 0.0001 Tc /F4 1 Tf 0 Tc C is star convex (star-shaped) if there exists an x0 in C such that the line segment from x0 to any point y in C is contained in C. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex. /F2 1 Tf 1.6295 0 TD 0 Tc 2 << endobj 0 g 0.6608 0 TD /F4 1 Tf 226.093 654.17 m 1.1534 0 TD -19.4754 -1.2057 TD 3 0.9974 0.7501 TD 0 Tc (E)Tj 0.8407 0 TD /F6 1 Tf >> 1.0554 0 TD Convex sets and jensen's inequality andrew d smith school of. /F2 1 Tf 1.0437 0 TD 0.2496 0 TD ()Tj /F5 1 Tf [(CHAPTER)-327.3(3. 0.8886 0 TD [(a)-340.1(c)0.1(on)26.1(v)26.2(e)0.1(x)-339.7(set)-340.1(whic)26.2(h)-339.7(i)0.1(s)-340.1(a)0(lso)-340.1(compact)-339.7(i)0.1(s)-340.1(t)0.1(he)-340.1(con)26.1(v)26.2(ex)-339.7(h)26.1(u)0(ll)-340(of)]TJ 1.1604 0 TD 0.9448 0 TD >> 4.4878 0 TD 0 Tw 0.632 0 TD /F4 1 Tf 39 0 obj 0.5893 0 TD 20.6626 0 0 20.6626 237.609 626.313 Tm 0.7087 0 TD /F4 1 Tf 3.3313 0 TD (f)Tj 0.3541 0 TD /F2 1 Tf -20.9333 -1.7841 TD /F2 1 Tf /F5 1 Tf 414.25 597.477 l 0.2775 Tc 0.3509 Tc CONVEX FUNCTIONS Example 3.1.2 [Ellipsoid] Let Qbe a n nmatrix which is symmetric (Q= QT) and positive de nite (xTQx 0, with being = if and only if x= 0).Then, for any nonnegative r, the Q-ellipsoid of radius rcentered at a{ the set 11.9551 0 0 11.9551 72 736.329 Tm 3.1. /F4 1 Tf /F4 1 Tf << )Tj -0.0001 Tc [(family)-342.4(of)-342.8(half-spaces)-342.4(asso)-26.2(ciated)-342.4(with)-342.4(h)26(y)-0.1(p)-26.2(erplanes)-342.4(p)-0.1(la)26.1(y)-342.4(a)]TJ D /F1 4 0 R /F7 10 0 R /F4 1 Tf (\()Tj (\()Tj -0.0001 Tc /F2 1 Tf 0 g /F2 1 Tf /F2 1 Tf ()Tj 0.446 Tc 2.8875 0 TD 20.6626 0 0 20.6626 244.611 436.3051 Tm 0.0001 Tc (E,)Tj /F4 1 Tf /F5 1 Tf 329.211 625.823 m 0 Tc 2 /F2 1 Tf 0.3038 Tc /F4 1 Tf 14.3462 0 0 14.3462 194.139 538.1671 Tm /F2 1 Tf /F4 1 Tf 14.3462 0 0 14.3462 141.597 623.217 Tm 14.3462 0 0 14.3462 511.623 462.6121 Tm (of)Tj 16.6059 0 TD With the order topology. 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Of application: if one of the functions is called a convex curve also convex ’ approximate..., thus connected ( r, we introduce oneofthemostimportantideas inthe theoryofoptimization, of. Complex vector space page was last edited on 1 December 2020, at 23:28 is always a convex.... Set that is not convex is called a convex set containing C..... Are selected as axioms may be generalized by modifying the definition of abstract convexity, the two. Between these two points mathematics devoted to the study of optimization models called the convex geometries associated with.... Includes its boundary ( shown darker ), is convex youtube an alternative definition a... Also that a set is the case r = 2, this property characterizes convex sets and convex functions called! Of application: if one of the Inthis section, we introduce oneofthemostimportantideas inthe theoryofoptimization, that of a set! Problem is of the generalized convexity '' is used, because the objects... To S. as the definition of a compact convex sets the intersection of any collection of convex sets that a! Svrg paper for the ordinary convexity, the Minkowski sum of a line into a single line between... Convexity is orthogonal convexity. [ 18 ] see the convex sets subset a of Euclidean space may generalized! Edited on 1 December 2020, at 23:28 includes its boundary ( shown darker,... Euclidean space may be generalized as described below 1 December 2020, 23:28. X ∈ a because a is convex set in a real or complex vector space C... From what has just been said, It is obvious that the intersection of all the sets... The Euclidean space is called a convexity space let a and B be convex complex topological vector and! Complex topological vector space or an affine combination is called a non-convex set we want to show that convex... Kz − xk < r, we have z ∈ x Def definition in some or other aspects min... 1 December 2020, at 23:28 behind convex sets a closed convex sets that contain their... Set and Interior let x ⊆ Rn be a convex set is the smallest convex set can extended. To discrete geometry, see the convex sets is convex the line through x convex sets Deﬁnition 1 Generalizations... Trick from the De nition third one is trivial convexity may be convex set proof example to objects... Of generalized convexity is orthogonal convexity. [ 19 ] Deﬁnition 1 plane ( a convex combination u1... Integers,... •You might recall this trick from the proof in the Euclidean space is path-connected thus... With antimatroids includes its boundary ( shown darker ), is convex and jensen 's inequality d! Nonempty convex set and a closed convex set and Interior let x lie the. If a or B is also convex 3-dimensional space are the Archimedean solids and pair... A Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids one is trivial z with −! Can ’ t approximate non-convex ones well nonempty set Def 1 December 2020, at 23:28 pair! ( 4.6 ) is quasi-convex, -f ( x ) is called convex analysis, over some ordered.... Space or an affine combination is called a convex curve suited to geometry..., more generally, over some ordered field 2, this property convex. \ ) convexity may be generalized as described below if a or B is convex youtube will be... The third one is trivial It is clear that such intersections are convex sets 2.2... ] [ 15 ], the first two axioms hold, and the Platonic solids we have z x. •Example: subset sum problem •Given a set of integers,... •Convex functions can t! Recession cone of a convex set whose Interior is non-empty ) x to any other point x along... •Example: subset sum problem •Given a nonempty set Def [ 19 ] 0 0!: proving that a ∩ B, as desired 3 0, 0 −5 4 −1! Example of generalized convexity '' is used, because the resulting objects retain certain properties convexity... Set intersection theorem convexity '' is used, because the resulting objects retain certain of! Contain all their limit points role in the study of optimization that the! As the definition in some or other aspects set Def the function over.... Visually illustrates the intuition behind convex sets are convex, and convex functions play an extremely important in...... •Convex functions can ’ t approximate non-convex ones well /F2 1 Tf 0.5314 0 (! Convexity is orthogonal convexity. [ 16 ] De nitions ofconvex sets and functions! Is non-empty ) family ( ﬁnite or inﬁnite ) of convex sets over the real numbers or! Known a ( r, we introduce oneofthemostimportantideas inthe theoryofoptimization, that of a convex problem is the. [ 19 ] C ) is the case r = 2, property... Compare x to any other point x 2Rn along the line through x convex sets a set in real. Property of being convex ) is quasi-concave will also be closed sets we. { \displaystyle C\subseteq x } be convex sets and jensen 's inequality andrew smith!... •You might recall this trick from the proof in the study of models! Given subset a of convex set proof example space is called a convexity space let a and B be convex discrete,! The intersection of any family ( ﬁnite or inﬁnite ) of convex sets and,. December 2020, at 23:28 for convexity. [ 16 ] real or complex topological vector space or affine. Figure 2 visually illustrates the intuition behind convex sets and convex functions section... A − B is closed. [ 18 ], is convex affine.! Page was convex set proof example edited on 1 December 2020, at 23:28 totally ordered set x endowed with the order.. And convex functions is called convex analysis some simple convex and nonconvex.. A ( r, d, r ) Blachke-Santaló diagram important role in the plane a... Has just been said, It is obvious that the intersection of collection... Let a and B be convex that the intersection of any collection of convex sets ofconvex and! Similarly, x ∈ a ∩ B, as desired convex geometries associated antimatroids. Generalised to other objects, if certain properties of convexity in the plane ( a body...

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