SELECTED TOPICS OF CONVEX AND DISCRETE GEOMETRY

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1 SELECTED TOPICS OF CONVEX AND DISCRETE GEOMETRY BERNARDO GONZÁLEZ MERINO 1. Classic radii in Euclidean space: diameter, inradius, minimal width Definition 1 (Diameter). Let K K n. The diameter of K is defined as D(K) = D(K, B n ) = max x y x,y K = max R({x, y}). x,y K Theorem (Jung, 1901). Let K K n. Then (1) R(K) n D(K) (n + 1). Moreover, equality holds if K is the n-dimensional regular simplex. Proof (Ball [1]). After a suitable translation and rescalation, we can suppose that K B n and R(K) = 1. By the Optimal Containment Condition, then p i K S n 1, λ i 0, i [m], m n + 1 s.t. m λ i p i = 0 and m λ i = 1. Technische Universität München, Germany address: Date: December 11, 016. The author is partially supported by Fundación Séneca, Science and Technology Agency of the Región de Murcia, through the Programa de Formación Postdoctoral de Personal Investigador, project reference 19769/PD/15, and the Programme in Support of Excellence Groups of the Región de Murcia, Spain, project reference 19901/GERM/15, and MINECO project reference MTM P, Spain. 1

2 GONZÁLEZ MERINO Indeed, we can assume that m = n + 1 (allowing some p i to be repeated). Hence () R(K) = 1 = λ i p i = 1 λ i p i + 1 λ j p j = 1 = 1 = 1 1 i,j 1 i,j 1 i,j, i j 1 D(K) λ i λ j i j j=1 λ i λ j p i λ i λ j p i p j 1 i,j λ i λ j p i p j = 1 (( D(K) λ i ) ) 1 (( D(K) λ i ) 1 n + 1 ( λ i ) λ i ) λ i λ j p i, p j i,j λ i λ j p j = 1 n n + 1 D(K), where the last inequality follows from Cauchy-Schwartz inequality. For the equality case, if R(K)/D(K) = n/((n + 1)), there is equality in all inequalities in (), and hence, there must exist p i K S n 1, i [n + 1], such that p i p j = D(K), for any 1 i < j n + 1, and thus, conv({p i : i [n + 1]}) is an n-dimensional regular simplex of diameter D(K) contained in K. Definition 3 (Inradius). Let K K n. The inradius of K is defined as r(k) = r(k, B n ) = max{ρ 0 : x + ρb n K, for some x R n } = max x K If x + r(k)b n K, we say that x + r(k)b n is the inball of K. Properties 4. The inball of K K n is, in general, not unique. min x y. y bd(k) Theorem 5 (Optimal Containment Condition). Let K K n be such that B n K. They are equivalent: (1) r(k) = 1. () There exist p 1,..., p m B n bd(k), m n + 1, such that 0 conv({p i : i [m]}). Theorem 6 (Steiner symmetrization). Let K K n, u = H L n n 1, for some u R n \{0}, and L := H. Let us define the Steiner symmetrization of K w.r.t. H to be Then: σ H (K) := x H {x + (1/)[p q, q p] : [p, q] = K (x + L)}.

3 SELECTED TOPICS OF CONVEX AND DISCRETE GEOMETRY 3 (1) σ H (K) K n. () σ H (K) =. (3) D(σ H (K)) D(K). (4) R(σ H (K)) R(K). Theorem 7 (Steiner 1838). Let K K n. Then there exists a sequence (u i ) i N S n 1 such that lim σ (u i i ) σ (u 1 ) (K) = x + B n B n, for some x R n. Theorem 8 (Isodiametric inequality (Bieberbach, 1915)). Let K K n. Then D(K) n B n D(B n ) n. Proof. By Theorem 7, letting (u i ) i N a sequence of directions such that and by Theorem 6 then lim σ (u m m ) σ (u 1 ) (K) = x + B n B n, σ (u m ) σ (u 1 ) (K) = and D(σ (u m ) σ (u 1 ) (K)) D(K), the continuity of and D( ) yields D(K) n lim σ (um ) σ (u 1 ) (K) m D(σ (u m ) σ = x + (/ B n )B n (u 1 ) (K))n D(x + (/ B n )B n ) n = B n D(B n ) n Definition 9. Let K K n. The minimal width of K is defined as w(k) = w(k, B n ) = min{h(k, u) + h(k, u) : u bd(b n )}. Remark 10. Let K K n. Then r(k) w(k) D(K) R(K). Theorem 11 (Blaschke, 1916). Let K K. Then w(k) r(k) 3, and equality holds iff K is an equilateral triangle. Theorem 1 (Pál problem (Pál, 1)). Let K K. Then where S is the equilateral triangle. w(k) S w(s ) = 1, 3 Proof. We can suppose without loss of generality that w(k) = 1. By Remark 10 then r(k) 1/. By Theorem 5, if x + r(k)b K, then the inball x + r(k)b touches the boundary of K either in two diametrical points of x+r(k)b, or in three points p i, i = 1,, 3, forming an acute triangle. In the first case, w(k) = r(k) and hence r(k) = 1/, from which π/4 > 1/ 3. In the second case, K is contained in a triangle S whose edges are contained in the three supporting lines of K and x + r(k)b at p i, i = 1,, 3. Moreover, due to w(k) = 1, each

4 4 GONZÁLEZ MERINO of the three regions of S outside x + r(k)b has to contain a point q i at distance 1 r(k) from x, i = 1,, 3. Hence conv({q i : i = 1,, 3} (x + r(k)b )). Calling r := r(k), it is easy to check that each of those three regions each has area r ( ) r 1 r r arccos 1 r (see Figure 1). Hence πr + 3 ( r ( )) r 1 r r arccos =: f(r). 1 r Since f(r) is an increasing function in r [0, 1/], and since Theorem 11 says that r w(k)/3 = 1/3, we conclude ( ) 1 f =

5 SELECTED TOPICS OF CONVEX AND DISCRETE GEOMETRY 5 references References [1] Keith Ball. An elementary introduction to modern convex geometry. Flavors of geometry, 31:1 58, 1997.

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