The set of prepolars

First of all, it is interesting to link the two extreme representations/prepolars introduced so far, and to confirm the intuition suggested by Figure 5.4:

Proposition 5.3.7. Appending0 to V•gives the standard polar: γV = max

 µV, 0

i.e. V◦ = conv V•_{∪ {0}}
= [0, 1]V•.

Proof. For r ∈ V∞,γV(r) = 0, while µV(r) 6 0 (Proposition 5.3.6). For r /∈ V∞, Lemma 5.3.2 givesγV(r) = µV(r) > 0 because γV andµV are two particular representations.

Altogether, the first equality holds. Its geometric counterpart is [103, Thm. C.3.3.2]; and because V•is convex compact, its closed convex hull with 0 is the sets ofαd + (1− α)0 for α ∈ [0, 1]. 

In summary, the set of representations – or of prepolars – is fully described as follows:

Theorem 5.3.8. The representations ofV (a closed convex neighborhood of the origin) are the finite- valued sublinear functionsρ satisfying

σV•= µ_V 6 ρ 6 γ_V = σ_V◦ = max{0, µ_V} . (5.24) Geometrically, the prepolars ofV , i.e. the sets G whose support function represents V , are the sets sandwiched between the two extreme prepolars ofV :

G◦= V _⇐⇒ V•_{⊂ conv (G) ⊂ V}◦= conv V•_{∪ {0}}
= [0, 1]V•.

Proof. In view of Corollary 5.3.3 and Propositions 5.3.6, 5.3.7, we just have to prove that aρ sat- isfying (5.24) does represent V . Indeed, if r _{∈ V then ρ(r) 6 γ}V(r) 6 1; if r /∈ V , then 1 < µV(r)6 ρ(r). The geometric counterpart is again standard calculus with support functions.  We end this section with a deeper study of prepolars, which will be useful in the sequel. The next result introduces the polar cone(V_∞)◦. WhenG is a cone, positive homogeneity can be used to replace the righthand side “1” in (5.19) by any positive number, or even by “0”: in particular,

V_∞◦ := (V_∞)◦ =_{{r ∈ R}q : σV∞(r)6 0} . (5.25) The notationV_∞◦ is used for simplicity, although it is somewhat informal; (V_∞)◦ and(V◦)_∞ differ, the latter is_{{0} since V}◦is bounded.

Lemma 5.3.9 (Additional properties of prepolars). With the notation (5.22), (5.23), (5.25), (i)V_∞◦ is the closure of dom σV,

(ii)R+Vb◦ =R+V• =R+V◦= dom σV.

Proof. First of all, letd /_{∈ V∞}◦: there isr _{∈ V∞}(R+r ∈ V ) and d>r > 0; then d>(tr) → +∞ for t→ +∞ and σV(d) cannot be finite, i.e. d /∈ dom σV. Thus,dom σV ⊂ V_∞◦; hencecl (dom σV)⊂ V_∞◦ becauseV_∞◦ is closed.

To prove the converse inclusion, taker /∈ (dom σV)◦: there isd such that σV(d) < +∞ and d>r > 0. Then d>(tr) _{→ +∞ when t → +∞; if r were in V∞}, thentr would lie in V and σV(d) would be+_{∞, a contradiction. Thus we have proved V∞⊂ (dom σ}V)◦. Taking polars and knowing thatdom σV is a cone,V_∞◦ ⊃ (dom σV)◦◦= cl (dom σV) (see [103, Prop. A.4.2.6]). This proves (i).

To prove (ii), observe first that bV◦ _{⊂ V}• _{⊂ V}◦ _{⊂ dom σ}V; and becausedom σV is a cone, R+Vb◦ ⊂ R+V• ⊂ R+V◦ ⊂ dom σV . (5.26) On the other hand, take06= d ∈ dom σV, so thatσV(d) > 0 by (5.21) and _σV1_(d)d∈ bV◦:d∈ R+Vb◦. Since0 also lies inR+Vb◦, we do havedom σV ⊂ R+Vb◦; (5.26) is actually a chain of equalities. To complete the proof, observe from Proposition 5.3.7 thatR+V◦ =R+V•. 

Conforti, Cornu´ejols, Daniilidis, Lemar´echal, Malick: Cut-generating functions

Mathematics of Operations Research xx(x), pp. xxx–xxx, c⃝200x INFORMS 13 Theorem 3.8 The representations of V (a closed convex neighborhood of the origin) are the finite-valued sublinear functions ρ satisfying

σV•= µV ! ρ ! γV = σV◦= max{0, µV} . (24) Geometrically, the prepolars of V , i.e. the sets G whose support function represents V , are the sets sandwiched between the two extreme prepolars of V :

G◦= V ⇐⇒ V•⊂ conv (G) ⊂ V◦= conv!V•∪ {0}"= [0, 1]V•.

Proof. In view of Corollary 3.3 and Propositions 3.6, 3.7, we just have to prove that a ρ satisfying (24) does represent V . Indeed, if r _{∈ V then ρ(r) ! γ}V(r)! 1; if r /∈ V , then 1 < µV(r)! ρ(r). The geometric counterpart is again standard calculus with support functions. " We end this section with a deeper study of prepolars, which will be useful in the sequel. The next result introduces the polar cone (V_∞)◦_{. When G is a cone, positive homogeneity can be used to replace} the righthand side “1” in (19) by any positive number, or even by “0”: in particular,

V_∞◦ := (V_∞)◦=_{{r ∈ R}q _{: σ}

V∞(r)! 0} . (25)

The notation V◦

∞ is used for simplicity, although it is somewhat informal; (V∞)◦ and (V◦)∞ differ, the latter is_{{0} since V}◦ _{is bounded.}

Lemma 3.9 (Additional properties of prepolars) With the notation (22), (23), (25), (i) V◦

∞ is the closure of dom σV, (ii)R+V#◦=R+V•=R+V◦ = dom σV.

Proof. First of all, let d /_{∈ V}◦

∞: there is r ∈ V∞ (R+r ∈ V ) and d⊤r > 0; then d⊤(tr) → +∞ for

t _{→ +∞ and σ}V(d) cannot be finite, i.e. d /∈ dom σV. Thus, dom σV ⊂ V_∞◦; hence cl (dom σV)⊂ V_∞◦ because V_∞◦ is closed.

To prove the converse inclusion, take r /∈ (dom σV)◦: there is d such that σV(d) < +∞ and d⊤r > 0. Then d⊤(tr)_{→ +∞ when t → +∞; if r were in V}∞, then tr would lie in V and σV(d) would be +∞, a contradiction. Thus we have proved V∞ ⊂ (dom σV)◦. Taking polars and knowing that dom σV is a cone, V◦

∞⊃ (dom σV)◦◦= cl (dom σV) (see [17, Prop. A.4.2.6]). This proves (i). To prove (ii), observe first that #V◦⊂ V•_{⊂ V}◦_{⊂ dom σ}

V; and because dom σV is a cone,

R+V#◦⊂ R+V•⊂ R+V◦⊂ dom σV . (26) On the other hand, take 0_{̸= d ∈ dom σ}V, so that σV(d) > 0 by (21) and _σV1_(d)d∈ #V◦: d∈ R+V#◦. Since

0 also lies in R+V#◦, we do have dom σV ⊂ R+V#◦; (26) is actually a chain of equalities. To complete the

proof, observe from Proposition 3.7 thatR+V◦=R+V•. "

φ v e P◦_{= bP}◦ 1 u d V = P r(d) ψ

Figure 5: Trouble appears if the neighborhood has no asymptote

Beware that really pathological prepolars can exist, Figure 5 illustrates a well-known situation. Its left part displays the parabolic neighborhood V = P _{⊂ R}2_{defined by the constraint ψ! 1−}1

2φ

2_{. A direction} d = (u, v) with v > 0 exposes the point r(d). When v _{↓ 0, the component of r(d) along d (namely φ)} goes to +∞, which does bring trouble. Computing r(d) is an exercise resulting in

σP(d) = σP(u, v) = ⎧ ⎨ ⎩ 0 if d = 0 , v +u_2v2 if v > 0 , +_∞ if v! 0 ; (27) two phenomena are then revealed.

Figure 5.5: Trouble appears if the neighborhood has no asymptote

Beware that really pathological prepolars can exist, Figure 5.5 illustrates a well-known situation. Its left part displays the parabolic neighborhoodV = P ⊂ R2_{defined by the constraintψ6 1 −}1

2φ2. A directiond = (u, v) with v > 0 exposes the point r(d). When v_{↓ 0, the component of r(d) along} d (namely φ) goes to +_{∞, which does bring trouble. Computing r(d) is an exercise resulting in}

σP(d) = σP(u, v) =    0 ifd = 0 , v +u_2v2 ifv > 0 , +∞ ifv6 0 ; (5.27)

two phenomena are then revealed. First, bV◦is defined by the equation

v + u 2

2v = 1 , i.e. 2(v

2_{− v) + u}2 _{= 0 .}

This is an ellipse passing through the origin (right part of Figure 5.5); yet 0 cannot lie in bV◦, since σP(0) = 06= 1. Thus, bP◦ is not closed and, more importantly,0∈ P•.

The second phenomenon is a violent discontinuity of σP at 0. In fact, fix α > 0 and let dk = α

k,k12

; thendk→ 0, while σP(dk)→ α 2

Both phenomena are due to (local) unboundedness ofσP on its domain, which is thus not closed; if(uk, vk) ∈ dom σP tends to any(u, 0) with u 6= 0, then σP(uk, vk) → +∞. Ruling out such a behaviour brings additional useful properties:

Corollary 5.3.10 (Safe prepolars). If0 /_{∈ V}•, then

R+Vb◦ =R+V• =R+V◦= dom σV = V_∞◦ (5.28)

andint V_{∞6= ∅ (the polar V∞}◦ is a so-called pointed cone). Proof. When 0 /∈ V•_{, R}

+V• is closed ([103, Prop. A.1.4.7]). Then apply Lemma 5.3.9: by (ii) dom σV is closed and (5.28) follows from (i).

Now we separate 0 fromV•: there is somer such that σV•(r) < 0. By continuity of the finite- valued convex functionσV•, this inequality is still valid in a neighborhood ofr: σV• 6 0 over some nonzero ballB around r. By Lemma 5.3.9(ii),

σV◦ ∞(d) = σR+V•(d) = sup t>0 sup d∈V•td >_{r = sup} t>0 tσV•(d) ,

so thatσV_∞◦ enjoys the same property: by (5.25), B is contained in (V_∞◦)◦. Proposition A.4.2.6 of

[103] finishes the proof. 

Property (5.28) means closedness of dom σV and is rather instrumental. We mention another simple assumption implying it:

Proposition 5.3.11. IfV = U + V_∞, whereU is bounded, then dom σV = V_∞◦.

Proof. The support function of a sum is easily seen to be the sum of support functions: σV = σU+ σV∞. Everyd∈ V∞◦ then satisfiesσV(d) = σU(d), a finite number when U is bounded.  Let us put this section in perspective. The traditional gauge theory defines via (5.16), (5.19) the polarity correspondenceV _{↔ V}◦ for compact convex neighborhoods of the origin. We generalize it to unbounded neighborhoods, whose standard gauge is replaced via Definition 5.2.6 by their family of representations. Each representation ρ, which may assume negative values, gives rise to ∂ρ(0) – which we call a prepolar ofV . Theorem 5.3.8 establishes the existence of a largest element (the usual polar V◦) and of a smallest element (V•) in the family of (closed convex) prepolars of V . Gauge theory is further generalized in [184], in which 0 may lie on the boundary ofV . Our stricter framework allows a finer analysis of the smallest prepolar; in particular, the property0 /_{∈ V}• helps avoiding nasty phenomena.