A multi-plank generalization of the Bang and Kadets inequalities
AA MULTI-PLANK GENERALIZATION OF THE BANG AND KADETSINEQUALITIES
ALEXEY BALITSKIY
Abstract.
If a convex body K ⊂ R n is covered by the union of convex bodies C , . . . , C N ,multiple subadditivity questions can be asked. Two classical results regard the subaddi-tivity of the width (the smallest distance between two parallel hyperplanes that sandwich K ) and the inradius (the largest radius of a ball contained in K ): the sum of the widthsof the C i is at least the width of K (this is the plank theorem of Thøger Bang), and thesum of the inradii of the C i is at least the inradius of K (this is due to Vladimir Kadets).We adapt the existing proofs of these results to prove a theorem on coverings bycertain generalized non-convex “multi-planks”. One corollary of this approach is a fam-ily of inequalities interpolating between Bang’s theorem and Kadets’s theorem. Othercorollaries include results reminiscent of the Davenport–Alexander problem, such as thefollowing: if an m -slice pizza cutter (that is, the union of m equiangular rays in the planewith the same endpoint) in applied N times to the unit disk, then there will be a pieceof the partition of inradius at least sin π/mN +sin π/m . Introduction
Let K be a convex set in R n endowed with the Euclidean norm. Two basic quantitiesmeasuring the “thickness” of K are its width w ( K ), the smallest distance between twoparallel hyperplanes that sandwich K , and its inradius r ( K ), the largest radius of a ballcontained in K . There are two classical results on the subadditivity of w ( · ) and r ( · ). Theorem 1.1 (Th. Bang [5]) . If a convex set K is covered by convex sets C , . . . , C N ,then N (cid:88) i =1 w ( C i ) ≥ w ( K ) . Theorem 1.2 (V. Kadets [16]) . If a convex set K is covered by convex sets C , . . . , C N ,then N (cid:88) i =1 r ( C i ) ≥ r ( K ) . If a convex set K sits inside an affine subspace L of R n , we use the notation r ( K ; L )for the inradius of K measured inside L . Definition 1.3.
Let 1 ≤ k ≤ n . The following quantities will be called the intrinsicinradii of a convex set K ⊂ R n .(1) The upper intrinsic inradius of K is defined as r ( k ) ( K ) = inf dim L = k r ( K | L ; L ) = inf dim L = k r ( K + L ⊥ ) , where L runs over the k -dimensional subspaces of R n , and K | L is the orthogonalprojection of K onto L . Supported in part by the Russian Foundation for Basic Research Grant 18-01-00036. a r X i v : . [ m a t h . M G ] S e p MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 2 (2) The lower intrinsic inradius of K is defined as r ( k ) ( K ) = inf dim L = k sup x ∈ L ⊥ r ( K ∩ ( L + x ); L + x ) , Equivalently, r ( k ) ( K ) can be defined via a Kakeya-type property: it is the supre-mum of numbers r such that the open ball of radius r of any k -dimensional sub-space can be placed in K after a translation.Those radii appeared (under different names) in multiple papers, e.g. [6, 7, 13, 14].Some other notions of successive radii (different from ours) in the context of certain plankproblems were considered in [9, 10, 11].Observe that r (1) ( K ) = r (1) ( K ) = w ( K ) / r ( n ) ( K ) = r ( n ) ( K ) = r ( K ). It is clearthat r ( k ) ( K ) ≥ r ( k ) ( K ), but in general it might happen that this inequality is strict; forinstance, this happens for the regular tetrahedron in R and k = 2.The following result, interpolating between Theorem 1.1 and Theorem 1.2, will followas a corollary of the main theorem, Theorem 3.1. Theorem 1.4.
If a convex set K is covered by convex sets C , . . . , C N , then for any ≤ k ≤ n , N (cid:88) i =1 r ( k ) ( C i ) ≥ r ( k ) ( K ) . Most often Theorem 1.1 is formulated in terms of coverings by planks; in this formu-lation it answered Tarski’s question [18]. A plank is the set of all points between twoparallel hyperplanes. We will interpret Theorem 1.4 in terms of coverings by certain non-convex “planks” (see Definition 2.1 and Figures 1, 2 for examples) and adapt classicalproofs of Theorems 1.1, 1.2 to give a one-page proof of a more general plank theorem(Theorem 3.1).Another type of corollaries that can be immediately deduced from the main theoremis akin to the Davenport–Alexander problem (see [3], where its relation to Bang’s plankproblem is explained), and a version of Conway’s fried potato problem (see [9], especiallyTheorem 2 therein). A corollary of Theorem 3.1 tells us that one can arbitrarily apply acommercial pizza cutter to one’s favorite pizza several times and still find a decently-sizedslice.
Theorem 1.5.
Let us call an m -fan ( m ≥ ) the union of m rays in the plane with thesame endpoint and with all angles πm . If the unit disk is partitioned by m -fans S , . . . , S N ,then there is a piece of inradius at least sin π/mN +sin π/m . In the case m = 2 it recovers a well-known Davenport-type result, which is equivalentto Tarski’s plank problem for disk.Section 2 introduces our notion of a multi-plank and gives several examples. The maintheorem in Section 3 is followed by the proofs of Theorem 1.4 and Theorem 1.5 (togetherwith its higher-dimensional generalizations). Section 4 establishes further properties ofmulti-planks with the hope to illustrate the concept and make Definition 2.1 less obscure.Section 5 discusses to what extent the main theorem generalizes to the case when R n is endowed with a non-Euclidean norm, whose unit ball need not be centrally symmet-ric, generally speaking. The normed counterparts of Theorems 1.1 and 1.2 are widelyopen questions. The former, known as Bang’s conjecture on relative widths, is solved byK. Ball [4] for the case when the unit ball is centrally symmetric. The latter is far lessunderstood, with some progress towards the case of partitions (instead of coverings) madein [2]. MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 3
Acknowledgements.
The most general multi-plank definition was hinted to me by Ro-man Karasev, whom I thank sincerely. I am also grateful to Alexandr Polyanskii for thefruitful discussions that we had. I thank Aleksei Pakharev for his help with the figures.Finally, I thank the referee, whose comments improved the exposition.2.
Multi-planks
Definition 2.1.
Let V = { v , . . . , v m } , m ≥
2, be a set of points in R n , such that theclosed ball with the smallest radius containing V is centered at the origin. Denote by r ( V ) its radius.(1) The set P = (cid:110) x ∈ R n (cid:12)(cid:12)(cid:12) ∀ j ∈ [ m ] ∃ j (cid:48) ∈ [ m ] such that | x | < | x − v j + v j (cid:48) | (cid:111) will be called the open centered multi-plank generated by V .(2) The closure P of P will be called the closed centered multi-plank generated by V .(3) A multi-plank generated by V is a translate of P or P .In all these cases, the radius r ( V ) will be called the inradius of a multi-plank (this wordchoice will be justified by Lemma 4.8). The dimension of the affine hull of V will be calledthe rank of a multi-plank. Example . If V = { u, − u } for 0 (cid:54) = u ∈ R n , then the corresponding rank 1 (opencentered) multi-plank is just the ordinary (open) plank P = (cid:8) x ∈ R n (cid:12)(cid:12) −| u | < (cid:104) x, u (cid:105) < | u | (cid:9) . Example . Let V = { v , . . . , v n +1 } ⊂ R n be a set of affinely independent vectors oflength r whose convex hull contains the origin in its interior. It follows that the smallestball containing V is B r , the ball of radius r centered at the origin. The correspondingrank n (open centered) multi-plank P can be described as follows. For each j ∈ [ n + 1],draw the tangent hyperplane H j to the ball B r at the point v j . Those hyperplanes bounda simplex ∆. Consider the union F of rays with the common endpoint at the origin thatintersect the ( n − F is the fan dividing spaceinto convex regions (in Section 4 it will be explained how these regions are related to the Vorono˘ı diagram of V ). The multi-plank P looks like a thickened fan F , with the widthsof its “wings” defined so that ∂P passes through each of the v j . (See Figure 1 for anexample.) Example . Let V = { v , . . . , v k +1 } ⊂ R n , 1 ≤ k ≤ n , be a set of vectors of length r whose convex hull is k -dimensional and contains the origin in its relative interior. Itfollows that the smallest ball containing V is B r , the ball of radius r centered at the origin.The corresponding rank k (open centered) multi-plank P is the Minkowski sum of the k -dimensional multi-plank generated by V in its affine hull (as in the previous example)with the orthogonal ( n − k )-dimensional subspace.The multi-planks as in the examples above (and their closures) will be called simple .An example of a non-simple multi-plank can be obtained, for instance, if one takes V consisting of three vectors whose endpoints form an obtuse triangle: v = − v and | v | < | v | . Another important family of non-simple multi-planks shows up in Proposition 4.1(see Figure 2). Lemma 2.5.
Let ≤ k ≤ n . Any open (closed) convex set C with finite intrinsic radius r ( k ) ( C ) can be placed inside an open (closed) simple multi-plank of rank at most k andinradius r ( k ) ( C ) . MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 4
Figure 1.
A simple multi-plank in the plane
Proof.
The quantity r ( C | L ; L ), where L is a k -dimensional subspace, depends on L ina lower semi-continuous way (in fact, it is continuous whenever finite). Therefore, wecan pick L delivering minimum in the definition r ( k ) ( C ) = inf dim L = k r ( C | L ; L ). Let c + B r be the largest k -ball in C | L ⊂ L . Next, there are points c + v , . . . , c + v m ∈ L (forsome 2 ≤ m ≤ k + 1) in the intersection of (relative) boundaries of C | L and of c + B r ,“certifying” that r was indeed maximal, that is, the following two conditions are satisfied.(C1) The set C lies in the intersection of halfspaces (cid:8) x ∈ R n (cid:12)(cid:12) (cid:104) x − c, v j (cid:105) < r (cid:9) , over j ∈ [ m ]. (In the case when C is closed, the inequalities are non-strict.)(C2) The point c is contained in the relative interior of conv { c + v , . . . , c + v m } .Let us show that the multi-plank c + P , where P is generated by V = { v , . . . , v m } ,as in Example 2.4, will do. By shifting everything, we can assume c coincides with theorigin. Suppose there exists a point x ∈ C \ P . By definition of the multi-plank P , thereis j ∈ [ m ] such that | x | ≥ (cid:12)(cid:12)(cid:12) x − v j + v j (cid:48) (cid:12)(cid:12)(cid:12) for all v j (cid:48) ∈ V. (In the closed case, the inequalities are strict.) Therefore | x | ≥ (cid:12)(cid:12)(cid:12) x − v j + v j (cid:48) (cid:12)(cid:12)(cid:12) = | x | + 2 r − (cid:10) x, v j (cid:11) + 2 (cid:68) x − v j , v j (cid:48) (cid:69) . By condition (C2) above it is possible to pick v j (cid:48) ∈ V such that (cid:104) x − v j , v j (cid:48) (cid:105) ≥
0, hencehaving (cid:10) x, v j (cid:11) ≥ r + (cid:68) x − v j , v j (cid:48) (cid:69) ≥ r , which contradicts condition (C1). (cid:3) MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 5 Multi-plank theorem
Now we are ready to state the main result. The proof follows closely the ideas of Bangand Kadets. Our exposition also makes use of a trick by Bogn´ar [12].Recall from Definition 2.1 that if a multi-plank P is generated by V , then r ( V ) denotesthe radius of the smallest closed ball containing V ; we call it the inradius of P , eventhough this word choice is not yet justified. Theorem 3.1.
If a convex set K ⊂ R n is covered by multi-planks P , . . . , P N of rank atmost k then N (cid:88) i =1 r ( V i ) ≥ r ( k ) ( K ) , where V i is the generating set for P i .Proof. Every closed multi-plank can be covered by an open one of almost the same inra-dius; so without loss of generality we assume that the multi-planks are open.It suffices to consider the case when K is bounded, i.e., K ⊂ B R for some R . If not,we apply theorem for K ∩ B R and pass to the limit R → ∞ ; here we use r ( k ) ( K ∩ B R ) → r ( k ) ( K ) as R → ∞ .First we reduce the problem to the case of centered multi-planks and then deal sepa-rately with the centered case (this strategy can be traced back to Bogn´ar [12]). Step 1.
We think of R n as a coordinate subspace H ⊂ R n +1 ; say, H = { ( x, ∈ R n +1 | x ∈ R n } . Now both the set K and the multi-planks P i sit inside R n +1 . Pick apoint O = (0 n , D ) very far from the origin; here 0 n is the origin of R n and D ∈ R is large.For each i , build the cylinder C i = ( P i ∩ B R ) + (cid:96) i , where (cid:96) i is the line passing through O and through the center of P i . Those cylinders cover the cone (cid:98) K = conv( K ∪ { O } ). Thereare two statements to check:(1) each C i can be covered by a multi-plank of the same rank as P i , centered at O ,and of inradius close to r ( V i ) (the proximity depends on D );(2) the intrinsic inradius r ( k ) (cid:16) (cid:98) K (cid:17) is close to r ( k ) ( K ; H ). (The notation r ( k ) ( · ; H ) is tospecify the ambient space where the intrinsic inradius is measured.)For the first one, notice that C i splits as the orthogonal product of (cid:96) i and of the set A ( D ) which is an affine copy of P i ∩ B R shrunk negligibly (as long as D is large) alongone direction. The reader can convince themselves that A ( D ) can be covered by a scaledcopy of P i ∩ B R with the homothety coefficient tending to 1 as D → ∞ . This explainsthe first statement.For the second claim, fix d ∈ R large enough so that r ( k ) ( K ; H ) = r ( k ) ( K + [0 , d ]); here K + [0 , d ] is a shorthand for K + [(0 n , , (0 n , d )] ⊂ R n +1 . Now observe that (cid:98) K ∩ ( K + [0 , d ])converges to K +[0 , d ] in the Hausdorff metric, as D → ∞ . One can check that the function r ( k ) ( · ) is Hausdorff continuous, so we can write r ( k ) ( K ; H ) ≥ r ( k ) (cid:16) (cid:98) K (cid:17) ≥ r ( k ) (cid:16) (cid:98) K ∩ ( K + [0 , d ]) (cid:17) −→ D →∞ r ( k ) ( K + [0 , d ]) = r ( k ) ( K ; H ) . Now, applying the theorem in the centered case, we get the desired inequality with asmall error term, which decays as D → ∞ . Step 2.
Now we can assume that all the P i are centered at the origin. The proofhere follows closely the ideas from original papers by Bang and Kadets with certain MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 6 simplifications. Assume the contrary to the statement of theorem: α = r ( k ) ( K ) N (cid:80) i =1 r ( V i ) > . We define the
Bang set X = (cid:40) N (cid:88) i =1 v j i i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ j i ≤ m i (cid:41) , where V i = { v i , . . . , v m i i } is the generating set of P i . The strategy of the proof is to showthat X can be covered by a translate of K (assuming the contrary to the statement oftheorem) but at the same time X does not fit into (cid:83) P i . Step 2.1.
The Bang set splits as the Minkowski sum of the generating sets of the multi-planks: X = V + . . . + V N . By the definition of the lower intrinsic radius, V i can becovered by a translate of r ( V i ) r ( k ) ( K ) K (where bar denotes closure), hence by a translate of αr ( V i ) r ( k ) ( K ) K . Therefore, for some translation vector s ∈ R n , X = V + . . . + V N ⊂ s + αr ( V ) r ( k ) ( K ) K + . . . + αr ( V N ) r ( k ) ( K ) K = s + K. Step 2.2.
Suppose X ⊂ s + (cid:83) P i , s ∈ R n . Consider the farthest from the origin point in X − s ; let it be x = − s + (cid:80) Ni =1 v j i i . Fix i and consider the family of vectors ( x − v j i i ) + v j (cid:48) i i ,over j (cid:48) i ∈ [ m i ]. Since x is the longest among them, it follows that x / ∈ P i . Repeating thisover all i , we get a contradiction. (cid:3) Proof of Theorem 1.4.
If a convex set K is covered by convex sets C , . . . , C N , then each C i can be covered by a simple closed multi-plank with inradius r ( k ) ( C i ) (see Lemma 2.5).Now Theorem 3.1 implies the desired inequality: N (cid:88) i =1 r ( k ) ( C i ) ≥ r ( k ) ( K ) . (cid:3) Proof of Theorem 1.5.
Let α F = πm . Suppose the contrary, and denote r < sin α F N +sin α F theradius of the largest disk inscribed in the partition by fans. Pick a number r between r and sin α F N +sin α F . Then the disk B − r is covered by the multi-planks P , . . . , P N , where P i is the r -neighborhood of S i . The inradius of each multi-plank equals r sin α F , so usingTheorem 3.1 one gets the inequality N r sin α F ≥ − r > NN + sin α F , contradicting the assumption r < sin α F N +sin α F . (cid:3) Theorem 1.5 can be generalized to higher dimensions in the evident way; the onlydifficulty is to write down the guaranteed inradius in terms of the class of “pizza cutters”.In the examples below, the cutter shape is given by a certain fan F , dividing R n intoconvex cones so that F cuts out in the unit sphere S n − a bunch of regions all having thesame inradius α F in the intrinsic sphere metric dist S n − ( · , · ).(1) One can define an m -fan F in R n as the union of m half-planes of dimension( n −
1) sharing the same boundary ( n − α F = πm . MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 7 (2) For every regular polytope C ⊂ R n centered at the origin, one can consider thefan F consisting of the rays from the origin passing through the ( n − C . The regions cut out by F in the unit sphere are all congruent since C isregular. For example, in the case of regular simplex, α F = arccos n .(3) For every Coxeter hyperplane arrangement A in R n (that is, the set of hyper-planes passing through the origin and generating a finite reflection group), onecan consider the fan F consisting of the hyperplanes of A . The regions cut outby F in the unit sphere are all congruent since the reflection group acts transi-tively on the Weyl chambers. For example, in the case of type A n reflection group, α F = arccos (cid:113) n − n ( n +1) .All these examples are subsumed by the following more general Davenport-type theo-rem, whose proof is literally the same as the one of Theorem 1.5. Theorem 3.2.
Let G be a finite subgroup of SO ( n ) , acting on the unit sphere, and let O be the G -orbit of any point from the unit sphere. The Vorono˘ı diagram of the set O givesrise to a fan F ⊂ R n as follows: by definition, x ∈ F if the function f ( y ) = | x − y | , y ∈ O ,attains its minimum in at least two orbit points. The regions cut out by F in the unitsphere are all congruent since G acts transitively on them, and their spherical inradius is α F = 12 min y (cid:54) = y ∈ O dist S n − ( y , y ) . Now, if one cuts the unit ball of R n by N congruent copies of F , there will be a piece ofinradius at least sin α F N +sin α F .Question . Let F be a sufficiently regular codimension 2 “fan”. For instance, F can bethe union of the rays from the origin passing through the ( n − n -simplex, n ≥
3. The cuts are given by N congruent copies of F placed arbitrarily in R n .What is the largest radius of an open ball lying in the unit ball and avoiding the cuts?We finish this section with a brief discussion of the optimality of the main theorem.Theorem 3.1 has many “asymptotic equality cases”, different from trivial equality caseswhen N = 1 or k = 1. For example, the unit disc in the plane can be covered bytwo multi-planks of radius r slightly greater than 1 /
2, each generated by N (cid:29) r . By picking N sufficiently large one can get r arbitrarily close to 1 /
2. A similar example shows that Theorem 1.5 is asymptotically sharpfor each fixed m and N → ∞ . On the other hand, all those non-trivial asymptotic equalitycases involve multi-planks that are not simple. Meanwhile, the proof of Theorem 1.4 onlyexploits simple multi-planks. Given that, it would be interesting to know how sharpTheorem 1.4 is when, say, K is not centrally symmetric and N >
Multi-plank stratification
This section is devoted to a complete description of how multi-planks actually look like.First, we show how multi-planks can be efficiently used for covering unions of conventionalplanks. This idea might be helpful for certain plank problems. Next, we introduce thelanguage of anti-Vorono˘ı diagrams , and their dual anti-Delaunay triangulations . RecallExample 2.3: a simple multi-plank looks like an inflated fan, dividing R n into unboundedconvex regions, which form the so-called Vorono˘ı diagram of V . An equivalent definition ofa multi-plank, introduced in this section, tells us that this is always the case: associated to V , there is a nice subdivision of R n into unbounded convex regions, such that its separatingset can be thickened in order to get the multi-plank generated by V . The main resultof this section, Theorem 4.5, roughly speaking, explains how this thickening is done. We MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 8 use it to justify the word “inradius” used in Definition 2.1. Remarks 4.2 and 4.7 discusspotential applications of the techniques of this section.
Proposition 4.1.
Let V be the Bang set of the family of planks P i = (cid:8) x ∈ R n (cid:12)(cid:12) −| u i | < (cid:104) x, u i (cid:105) < | u i | (cid:9) ; that is, V consists of all combinations (cid:80) i ± u i , over all possible sign choices. Then theopen centered multi-plank P generated by V contains the union (cid:83) i P i (see Figure 2).Proof. Assume x / ∈ P ; then for a certain v j = (cid:80) i ε i u i , ε i ∈ { +1 , − } , we have | x | ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( x − v j ) + (cid:88) i ε (cid:48) i u i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for all ε (cid:48) i ∈ { +1 , − } . Therefore, | x | ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + (cid:88) i ( ε (cid:48) i − ε i ) u i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for all ε (cid:48) i ∈ { +1 , − } . Set all ε (cid:48) i equal to the corresponding ε i except one; then we get | x | ≥ | x − ε i u i | = | x | − ε i (cid:104) x, u i (cid:105) + 4 | u i | , for all i. This last line implies that x / ∈ P i , for each i ; hence, (cid:83) i P i ⊆ P . (cid:3) Figure 2.
A multi-plank covering two planks
MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 9
Remark . Notice that two non-parallel planks, as in Figure 2, have total half-width | u | + | u | , but their union is covered (even strictly covered unless (cid:104) u , u (cid:105) = 0) by amulti-plank of inradius max( | u + u | , | u − u | ), which is strictly less than | u | + | u | .Informally speaking, this gives a more “economical” way of measuring “total width” inthe following sense. Assume a symmetric convex set K (for instance, the unit ball) iscovered by planks; then some non-parallel pairs (or concurrent triples, etc.) are replacedby larger multi-planks, as in Proposition 4.1. For the resulting covering, the inequalitygiven by Theorem 3.1 is stronger than the one given by Theorem 1.1 for the originalcovering. It is interesting whether this approach can give anything non-trivial for any ofthe open plank problems, e.g., Andr´as Bezdek’s conjecture [8] on covering an annulus: ifthe unit disk with a small puncture is covered by planks, then their total width is at least2. Now we introduce an equivalent way to define a multi-plank using the language ofVorono˘ı diagrams. Definition 4.3.
Given a set V = { v , . . . , v m } of points in R n , the anti-Vorono˘ı diagram (or the farthest-point Vorono˘ı diagram ) is the partition R n = (cid:83) j ∈ [ m ] A jV , where the closed cells A V , . . . , A mV are given by A jV = (cid:110) x ∈ R n (cid:12)(cid:12)(cid:12) | x − v j | ≥ | x − v j (cid:48) | ∀ j (cid:48) ∈ [ m ] (cid:111) . In other words, A jV consists of all points for which the farthest element of V is v j .One should notice that all regions A jV are convex. Additionally, each cell A jV is eitherunbounded (if v j is an extreme point of conv V ) or empty (otherwise). The unboundednessof non-empty cells follows from the following simple claim: If x ∈ A jV then the entire ray { x + t ( x − v j ) | t ≥ } lies in A jV . In the sequel we use the notation A j − V for the anti-Vorono˘ıcell of the set − V = {− v , . . . , − v m } corresponding to the point − v j ; that is, A j − V = (cid:110) x ∈ R n (cid:12)(cid:12)(cid:12) | x + v j | ≥ | x + v j (cid:48) | ∀ j (cid:48) ∈ [ m ] (cid:111) . We are ready to rephrase Definition 2.1. Let V = { v , . . . , v m } , m ≥
2, be a set ofpoints in R n , such that the smallest ball containing V is centered at the origin. Then theset P = R n \ (cid:91) j ∈ [ m ] (cid:0) v j + A j − V (cid:1) is precisely the open centered multi-plank generated by V . Definition 4.4.
Given a set V = { v , . . . , v m } of points in R n , whose affine hull is theentire R n , an anti-Delaunay triangulation (or a farthest-point Delaunay triangulation ) isa triangulation of conv V satisfying the full sphere property : for each simplex of the trian-gulation, the (closed) ball whose boundary passes through the simplex vertices containsthe entire V .It is known that an anti-Delaunay triangulation always exists (see, e.g., [1, Section 4]),and is unique provided that no n + 2 points lie on a sphere. In the case when the affinehull of V is smaller than R n , one can define an anti-Delaunay triangulation inside theaffine hull of V .Now we give a finer description what a multi-plank looks like. We use the notation N T ( x ) for the cone of outer normals of a convex body T at a boundary point x ∈ ∂T ; bydefinition, N T ( x ) = { ν ∈ R n | (cid:104) ν, y − x (cid:105) ≤ ∀ y ∈ T } . If F is a face of T , we write N T ( F )for the cone of outer normals at any point from the relative interior of F . MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 10
Let V = { v , . . . , v m } ⊂ R n be the generating set of an open centered multi-plank P .If the rank of P is smaller than n , the multi-plank looks like the orthogonal product of asubspace and a lower-dimensional multi-plank. For this reason, we restrict our attentionto full-rank multi-planks for now.Consider an anti-Delaunay triangulation Σ of conv V regarded as a simplicial complex.For each top-dimensional cell σ of Σ, let S σ be the translated copy of σ such that theorigin is equidistant from the vertices of S σ . The simplices S σ do not overlap: this followsfrom the full sphere property of Σ. Indeed, if σ and σ are two anti-Delaunay cells, theyneed to be pushed apart in order to make their circumspheres concentric. (Here and belowby “circumsphere” we mean the sphere passing through all the vertices of a simplex, and“circumradius” refers to its radius. Note that this is not standard.)Let τ be a cell in Σ of dimension greater that 0. For each top-dimensional cell σ containing τ , find the corresponding face T τ,σ of S σ (the one that is a translated copy of τ ). In particular, T σ,σ = S σ . Consider the following set:( (cid:63) ) P τ = (cid:92) σ ⊃ τ (rint T τ,σ + N S σ ( T τ,σ )) , where the intersection is taken over all top-dimensional cells σ containing τ . For top-dimensional cells, this definition gives P σ = int S σ . We prove that the multi-plank P can be decomposed into strata P τ . Theorem 4.5.
Let P be a multi-plank of full rank generated by V ⊂ R n . Let Σ be an anti-Delaunay triangulation of V , viewed as a simplicial complex. For each top-dimensionalcell σ of Σ , let S σ be the translated copy of σ such that the origin is equidistant from thevertices of S σ . For each pair of cells τ ⊂ σ with < dim τ ≤ dim σ = n , let T τ,σ be theface of S σ that is a translated copy of τ . Let P τ be defined as in ( (cid:63) ) .With this notation, the multi-plank P admits the following stratification: P = n (cid:91) d =1 (cid:91) dim τ = d P τ , where the inner union is taken over all cells of Σ of dimension d . Note that we do not assume any genericity of V , except for its affine rank; but inthe special case when no n + 2 points of V lie on the same sphere, this description tellsus that locally, near each simplex S σ = conv { u , . . . , u n } , the multi-plank P looks like R n \ n (cid:83) j =0 ( u j + N S σ ( u j )) (see Figure 3). Proof of Theorem 4.5.
To begin with, we extend the definition of P τ to the case when τ = v j is a vertex in Σ. As before, for each top-dimensional cell σ containing v j , find thecorresponding vertex T v j ,σ of S σ . The stratum corresponding to v j is defined as P v j = (cid:92) σ (cid:51) v j (cid:0) T v j ,σ + N S σ ( T v j ,σ ) (cid:1) , where the intersection is taken over all top-dimensional cells σ containing v j . We claimthat P v j is nothing else as v j + A j − V , the shifted anti-Vorono˘ı cell from the alternativedefinition of a multi-plank. This is somewhat tedious but straightforward. The set v j + A j − V is defined by the system of inequalities | x | ≥ | x − v j + v j (cid:48) | , over v j (cid:48) ∈ V . In fact, onlythe vertices v j (cid:48) adjacent to v j in Σ contribute to this system; this is essentially the duality MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 11
Figure 3.
The stratification of a multi-plankbetween the anti-Delaunay triangulation and the anti-Vorono˘ı diagram. Equivalently, onecan write those inequalities as( ♠ ) (cid:28) x + v j (cid:48) − v j , v j (cid:48) − v j (cid:29) ≤ , v j (cid:48) adjacent to v j . On the other hand, each set T v j ,σ + N S σ ( T v j ,σ ) is defined by inequalities of the form( ♥ ) (cid:10) x − T v j ,σ , T v j (cid:48) ,σ − T v j ,σ (cid:11) ≤ , v j (cid:48) ∈ σ. Varying σ (cid:51) v j here, one gets those inequalities for all v j (cid:48) adjacent to v j in Σ.Observe that v j (cid:48) − v j = T v j (cid:48) ,σ − T v j ,σ , if v j and v j (cid:48) form an edge in σ . Next, T v j ,σ = T v j ,σ + T v j (cid:48) ,σ (cid:124) (cid:123)(cid:122) (cid:125) orthogonal to v j (cid:48) − v j + T v j ,σ − T v j (cid:48) ,σ (cid:124) (cid:123)(cid:122) (cid:125) = vj − vj (cid:48) . MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 12
Therefore, v j − v j (cid:48) and T v j ,σ lie in the same hyperplane orthogonal to v j (cid:48) − v j ; this provesthat inequalities ( ♠ ) and ( ♥ ) are equivalent, so P v j = v j + A j − V .To finish the proof, it suffices to observe that R n is the disjoint union of the strata P τ , over faces τ in Σ of any dimension. Indeed, for any point x ∈ R n we can considerthe nearest to x point y ∈ (cid:83) σ S σ (the union is over top-dimensional cells of Σ). If y ∈ rint T ( τ, σ ) then it is easy to see that x ∈ P τ (and that P τ is the only stratum containing x ). (cid:3) Remark . Theorem 4.5 remains true in the case when the rank of P ⊂ R n is less than n . In this case, the anti-Delaunay triangulation of V should be considered inside theaffine hull L of V , and in the definition of stratum ( (cid:63) ) the normal cone N S σ ( T τ,σ ) getsdecomposed as the Minkowski sum of the normal cone in L with L ⊥ . Remark . If P is a centered rank k multi-plank generated by V , the stratification of P is defined using the k -dimensional anti-Delaunay triangulation of V . Let ρ be the smallestcircumradius of a top-dimensional cell of that triangulation. Clearly, ρ ≥ r ( V ), and itmight happen that ρ > r ( V ), if not all vertices of conv V lie on the sphere of radius r ( V ).A direct corollary of Theorem 4.5 is that inside the ball B ρ of radius ρ the multi-plank P can be simplified; namely, P ∩ B ρ = P (cid:48) ∩ B ρ , where the multi-plank P (cid:48) is generatedby the subset of V consisting of vectors of length r ( V ). It would be interesting to knowwhether the proof of Jiang and Polyanskii [15] of L. Fejes T´oth’s zone conjecture can beretold using this trick in the language of multi-planks. We use Theorem 4.5 to justify the word “inradius” used in Definition 2.1. Notice thathere we refer to the intrinsic inradii of a possibly non-convex set; these are defined exactlyas in Definition 1.3.
Lemma 4.8.
Let P be an open multi-plank of rank k generated by V ⊂ R n . The radius r ( V ) (as in Definition 2.1) is indeed the inradius of P ; moreover, the upper intrinsic in-radii r ( k ) ( P ) , . . . , r ( n ) ( P ) , and the lower intrinsic radii r ( k ) ( P ) , . . . , r ( n ) ( P ) all equal r ( V ) .Proof. We can assume that P is centered. First we show that the open ball B r of radius r = r ( V ) is contained in P .Suppose x / ∈ P , then x ∈ v j + A j − V for some j . It means that the farthest from x − v j element of − V is − v j , that is, (cid:12)(cid:12) ( x − v j ) + v j (cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12) ( x − v j ) + v j (cid:48) (cid:12)(cid:12)(cid:12) , for all v j (cid:48) ∈ V. Therefore, | x | ≥ (cid:12)(cid:12)(cid:12) x − ( v j − v j (cid:48) ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) x − v j (cid:12)(cid:12) + 2 (cid:68) x − v j , v j (cid:48) (cid:69) + (cid:12)(cid:12)(cid:12) v j (cid:48) (cid:12)(cid:12)(cid:12) . It is possible to pick v j (cid:48) ∈ ∂B r such that (cid:104) x − v j , v j (cid:48) (cid:105) ≥
0, since B r is the smallest ballcontaining V . For such a choice of v j (cid:48) one gets | x | ≥ (cid:12)(cid:12)(cid:12) v j (cid:48) (cid:12)(cid:12)(cid:12) = r , thus proving that x / ∈ B r . While this paper was under review, Polyanskii released a preprint [17] proving even stronger versionof L. Fejes T´oth’s zone conjecture. The proof is partially inspired by certain multi-plank-related intuition,but there are other crucial ideas as well.
MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 13
We have shown that r ( n ) ( P ) ≥ r ( V ). Now we need to show that r ( k ) ( P ) ≤ r ( V ). Infact, it suffices to show that r ( n ) ( P ) ≤ r ( V ), since a rank k multi-plank is the Minkowskisum of a k -dimensional multi-plank with the orthogonal subspace.Let c + B ρ be the largest open ball contained in P , ρ = r ( n ) ( P ). Let its center belongto the stratum P τ of P , where τ = conv { u , . . . , u d } is a d -dimensional cell of the anti-Delaunay triangulation of conv V . The stratum P τ can be represented as P τ = s + rint τ + R , where s ∈ R n is a translation vector, and R is a certain closed set of dimension n − d ,orthogonal to τ . From the stratification result, Theorem 4.5, one can deduce that the sets s + u j + R are all disjoint from P . Hence, the radius ρ does not exceed the shortest amongthe distances dist( c, s + u j + R ) = | π τ ( c − s ) − u j | , where π τ ( c − s ) ∈ rint τ is the orthogonalprojection of c − s onto the affine hull of τ . If ρ > r = r ( V ), then the vertices of τ are allin B r \ ( π τ ( c − s ) + B ρ ), which can be strictly separated from π τ ( c − s ) by a hyperplane;this contradicts the fact π τ ( c − s ) ∈ rint τ . Therefore, ρ = r ( n ) ( P ) ≤ r ( V ). (cid:3) Multi-planks in normed spaces
The scheme of the proof of Theorem 3.1 can be repeated to an extent in the setting ofa normed space. Let R n be endowed with a (possibly, asymmetric) norm (cid:107) · (cid:107) whose openunit ball is B , an open bounded convex set containing the origin: (cid:107) x (cid:107) = inf { r | x ∈ rB } . We do not require B to be centrally symmetric, so in general (cid:107) x (cid:107) (cid:54) = (cid:107)− x (cid:107) (but the triangleinequality holds). Definition 5.1.
Let K be a convex set in an asymmetric normed space R n with the unitball B . Let 1 ≤ k ≤ n .(1) The upper intrinsic inradius r ( k ) B ( K ) is defined as the largest number r such that,for any codimension k subspace N , the homothet rB can be translated into K + N .(2) The lower intrinsic inradius r B ( k ) ( K ) is defined as the largest number r such thatany k -dimensional section (passing through the origin) of rB can be translatedinto K . Definition 5.2.
Let V = { v , . . . , v m } , m ≥
2, be a set of points in R n , such that (cid:107) v j (cid:107) ≤ r for all j , and V cannot be covered by a homothet of B smaller than rB . Definethe anti-Vorono˘ı cells as A j − V = (cid:110) x ∈ R n (cid:12)(cid:12)(cid:12) (cid:107) x + v j (cid:107) ≥ (cid:107) x + v j (cid:48) (cid:107) ∀ j (cid:48) ∈ [ m ] (cid:111) , and the open centered multi-plank generated by V as P = R n \ (cid:91) j ∈ [ m ] (cid:0) v j + A j − V (cid:1) . The number r = r B ( P ) is called the inradius of P , and the dimension of the convexhull of V is called the rank of P .We remark that the cells A j − V are no longer convex. See Figure 4 for an example of arank 1 plank in an asymmetric norm. In this figure, the unit ball of the norm is depictedin the middle (the origin is marked with a ‘+’ sign), the generating set is V = { v , v } .The proof of step 1 in Theorem 3.1 falls through in the normed case. It is no longer truethat a shifted multi-plank looks similar to the section of a higher-dimensional multi-plank;this is the reason why we have to consider only centered multi-planks. The proof of step2 is still valid, though, which gives us the following result. MULTI-PLANK GENERALIZATION OF THE BANG AND KADETS INEQUALITIES 14
Figure 4.
A multi-plank in a normed plane
Theorem 5.3.
If a convex set K in an asymmetric normed space R n with the unit ball B is covered by (centered) multi-planks P , . . . , P N of rank at most k , then N (cid:88) i =1 r B ( P i ) ≥ r B ( k ) ( K ) . In the case k = 1, K = B , Theorem 5.2 can be viewed as a result on the subadditivityof relative widths. Take a look at the “bent” plank P in Figure 4: it has the same lengthintersection with every line parallel to v − v . In this sense, P has a well-defined “relativewidth” r B ( P ) in this direction. In these terms, Theorem 5.2 says that if K covered by“bent” centered planks then the sum of their “relative widths” is at least 1. This mightbe reminiscent of Bang’s conjectured inequality on the sum of relative widths: if an openbounded convex set K containing the origin is covered by (conventional straight) planks P , . . . , P N , then N (cid:88) i =1 r (1) K ( P i ) ≥ . We finish with a strong conjecture subsuming Bang’s conjecture as well as many othersubadditivity statements.
Conjecture 5.4.
Let B be an open bounded convex set containing the origin, and let aconvex set K be covered by convex sets C , . . . , C N . Then for any ≤ k ≤ n , N (cid:88) i =1 r ( k ) B ( C i ) ≥ r B ( k ) ( K ) . References [1] A. Akopyan, A. Glazyrin, O. R. Musin, and A. Tarasov. The extremal spheres theorem.
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