Optimizing volume with prescribed diameter or minimum width

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1 Optimizing volume with prescribed diameter or minimum width B. González Merino* (joint with T. Jahn, A. Polyanskii, M. Schymura, and G. Wachsmuth) Berlin *Author partially funded by Fundación Séneca, proyect 19901/GERM/15, and by MINECO, project MTM P, Spain. Departament of Mathematical Analysis, University of Sevilla, Spain Einstein Workshop Discrete Geometry and Topology, FU Berlin, Germany, π-day 2018.

2 Kakeya problem (1917): Which region D minimizes area for which a needle of length 1 can be rotated inside D by 2π radians?

3 Kakeya problem (1917): Which region D minimizes area for which a needle of length 1 can be rotated inside D by 2π radians? Deltoid? No!

4 Kakeya problem (1917): Which region D minimizes area for which a needle of length 1 can be rotated inside D by 2π radians? Deltoid? No! Besicovitch (1919)

5 Kakeya problem (1917): Which region D minimizes area for which a needle of length 1 can be rotated inside D by 2π radians? Deltoid? No! Besicovitch (1919) Arbitrary small area

6 Convex Kakeya problem: Which convex region K minimizes area for which the breadth in each direction is at least 1?

7 Convex Kakeya problem: Which convex region K minimizes area for which the breadth in each direction is at least 1?

8 Convex Kakeya problem: Which convex region K minimizes area for which the breadth in each direction is at least 1? w(k) = min u w(k, u)

9 Convex Kakeya problem: Which convex region K minimizes area for which w(k) 1? w(k) = min u w(k, u)

10 Theorem (Pál, 1921) Let K be a planar convex set. Then A(K) w(k)

11 Theorem (Pál, 1921) Let K be a planar convex set. Then A(K) w(k) Equality holds iff K is an equilateral triangle.

12 Pál s problem Let K K n. Find K 0 K n and C n > 0 s.t. vol(k) w(k) n vol(k 0) w(k 0 ) n = C n.

13 Pál s problem Let K K n. Find K 0 K n and C n > 0 s.t. vol(k) w(k) n vol(k 0) w(k 0 ) n = C n. Observation: K 0 is w-minimal under inclusion.

14 Pál s problem Let K K n. Find K 0 K n and C n > 0 s.t. vol(k) w(k) n vol(k 0) w(k 0 ) n = C n. Definition (Reduced set): K is w-minimal under inclusion.

17 Lassak s question (1990): Does it exist reduced polytopes in dimension n 3?

18 Lassak s question (1990): Does it exist reduced polytopes in dimension n 3? Theorem (Martini, Swanepoel 04, Averkov, Martini 08) Let P K n be a polytope in n 3. If...

19 Lassak s question (1990): Does it exist reduced polytopes in dimension n 3? Theorem (Martini, Swanepoel 04, Averkov, Martini 08) Let P K n be a polytope in n 3. If... 1 it is a simplex, then P is not reduced.

20 Lassak s question (1990): Does it exist reduced polytopes in dimension n 3? Theorem (Martini, Swanepoel 04, Averkov, Martini 08) Let P K n be a polytope in n 3. If... 1 it is a simplex, then... 2 it is a pyramid, then P is not reduced.

21 Lassak s question (1990): Does it exist reduced polytopes in dimension n 3? Theorem (Martini, Swanepoel 04, Averkov, Martini 08) Let P K n be a polytope in n 3. If... 1 it is a simplex, then... 2 it is a pyramid, then... 3 it has n + 2 facets or n + 2 vertices, then P is not reduced.

22 Theorem 1 (G.M., Jahn, Polyanskii, Wachsmuth 17) Figure: A reduced polytope with 12 vertices and 16 facets

23 Question (f-vector) If P K n is a reduced polytope, find best c(n), C(n) > 0 s.t. f 0 (P) + c(n) f n 1 (P) f 0 (P) + C(n) (c(n), C(3) 4 0).

24 Isodiametric inequality (Bieberbach 1915) Let K K n. Then Equality holds iff K = B n 2. vol(k) D(K) n vol(bn 2 ) D(B n = 2 )n 2 n κ n.

25 Reverse isodiametric? If K = [ a, a] [ a 1, a 1 ], for a > 0 arbitrarily large, then A(K) D(K) 2 = 4 4 ( a a 2 ) 0.

26 Reverse isodiametric? If K = [ a, a] [ a 1, a 1 ], for a > 0 arbitrarily large, then Idea: Affine Geometry A(K) D(K) 2 = 4 4 ( a ) 0. a 2

27 Definition (Isodiametric position, Behrend 37, G.M., Schymura 18+) K K n is in isodiametric position if vol(k) D(K) n = sup vol(a(k)) A GL(n,R) D(A(K)) n.

28 Definition (Isodiametric position, Behrend 37, G.M., Schymura 18+) K K n is in isodiametric position if Definition vol(k) D(K) n = sup vol(a(k)) A GL(n,R) D(A(K)) n. V S n 1 is the set of diameters of K, and fulfills v V iff x K s.t. x + D(K)[0, v] K.

29 Theorem 2 (G.M., Schymura 18+) Let K K n be in IDP and V be its set of diameters. Then for every L i-dimensional subspace, i [n 1], ( ) i v V s.t. (L, v) arccos. n

30 Theorem 2 (G.M., Schymura 18+) Let K K n be in IDP and V be its set of diameters. Then for every L i-dimensional subspace, i [n 1], ( ) i v V s.t. (L, v) arccos. n

31 Theorem 2 (G.M., Schymura 18+) Let K K n be in IDP and V be its set of diameters. Then for every L i-dimensional subspace, i [n 1], ( ) i v V s.t. (L, v) arccos. n Sharpness? Identifying WHEN K is in isodiametric position!

32 Löwner position K B n 2 is in Löwner position if Bn 2 volume containing K. is the ellipsoid of minimum

33 Löwner position K B n 2 is in Löwner position if Bn 2 volume containing K. is the ellipsoid of minimum Theorem (John 1948, Ball 1992) Let K K n be 0-symmetric s.t. K B n 2. The following are equivalent: K is in Löwner position. There exist u i K S n 1, λ i 0, i [m], n m ( ) n+1 2, s.t. m λ i u i ui T = I n. i=1

34 Theorem 3 (G.M., Schymura 18+) Let K K n. The following are equivalent:

35 Theorem 3 (G.M., Schymura 18+) Let K K n. The following are equivalent: K is in isodiametric position.

36 Theorem 3 (G.M., Schymura 18+) Let K K n. The following are equivalent: K is in isodiametric position. 1 D(K) (K K) is in Löwner position.

37 Remark 1 (Dvoretzky-Rogers Factorization 1950) Let u i S n 1, λ i 0, s.t. I n = m i=1 λ iu i ui T. Then there exist 1 i 1 < < i n m s.t. ( ) j (L j, u ij+1 ) arccos where L j = span(u i1,..., u ij ). n

38 Remark 1 (Dvoretzky-Rogers Factorization 1950) Let u i S n 1, λ i 0, s.t. I n = m i=1 λ iu i ui T. Then there exist 1 i 1 < < i n m s.t. ( ) j (L j, u ij+1 ) arccos where L j = span(u i1,..., u ij ). n Remark 2 The vectors u i = 1 n (±1,..., ±1), i [2 n ] and L j = span(e 1,..., e j ) shows that Theorem 2 is best possible.

39 Theorem 4 (G.M., Schymura 18+) Let u i S n 1, λ i 0, i [m], with n m ( n+1 2 I n = m λ i u i ui T. i=1 ), be such that Then min 1 i<j m ut i u j ( n ( 1 ( 2) m ) 2. m 2) n The inequality is sharp for m = n, n + 1, and sometimes for ( ) n+1 2.

40 Theorem 4 (G.M., Schymura 18+) Let u i S n 1, λ i 0, i [m], with n m ( n+1 2 I n = m λ i u i ui T. i=1 ), be such that Then min 1 i<j m ut i u j ( n ( 1 ( 2) m ) 2. m 2) n The inequality is sharp for m = n, n + 1, and sometimes for ( ) n+1 2. (u i1, u i2 ) ( ) arccos n 1 ( ) arccos 1 n+2 DR-F GMS

41 Lemma 1 (Gen. Cauchy-Binet formula) Let A R n m, B R m n, and I, J ( [n]) i. Then det((ab) I,J ) = det(a I,P ) det(b P,J ). P ( [m] i )

42 Lemma 2 Let u i S n 1, λ i 0, i [m], n m ( ) n+1 2, be s.t. I n = m i=1 λ iu i ui T. Then for every i [n] ( ) n = λ J det((u J ) T U J ), i J ( [m] i ) where λ J = j J λ j and U J = (u j : j J).

43 Lemma 2 Let u i S n 1, λ i 0, i [m], n m ( ) n+1 2, be s.t. I n = m i=1 λ iu i ui T. Then for every i [n] ( ) n = λ J det((u J ) T U J ), i J ( [m] i ) where λ J = j J λ j and U J = (u j : j J). For instance, n = m i=1 λ i.

44 Lemma 3 Let m N, m 2, λ i 0, i [m], and c 0 be such that c = m i=1 λ i. If f (λ 1,..., λ m ) = λ i λ j, then 1 i<j m max f (λ 1,..., λ m ) = f (c/m,..., c/m) = c2 (m 1). λ i 0 2m

45 Theorem 4 (G.M., Schymura 18+) Let u i S n 1, λ i 0, i [m], with n m ( n+1 2 I n = m λ i u i ui T. i=1 ), be such that Then min 1 i<j m ut i u j ( n ( 1 ( 2) m ) 2. m 2) n The inequality is sharp for m = n, n + 1, and sometimes for ( ) n+1 2.

46 Proof. ( ) n = ( 1 u T λ i λ j det i u j 2 u T 1 i<j m i u j 1 ) [Lem. 2]

47 Proof. ( ) n = ( 1 u T λ i λ j det i u j 2 u T 1 i<j m i u j 1 = λ i λ j (1 (ui T u j ) 2 ) 1 i<j m ) [Lem. 2]

48 Proof. ( ) n = ( 1 u T λ i λ j det i u j 2 u T 1 i<j m i u j 1 = λ i λ j (1 (ui T u j ) 2 ) 1 i<j m max λk =n, λ k 0 1 i<j m ) [Lem. 2] λ i λ j max (1 1 i<j m (ut i u j ) 2 )

49 Proof. ( ) n = ( 1 u T λ i λ j det i u j 2 u T 1 i<j m i u j 1 = λ i λ j (1 (ui T u j ) 2 ) 1 i<j m ) [Lem. 2] max λ i λ j max (1 λk =n, λ k 0 1 i<j m (ut i u j ) 2 ) 1 i<j m ( n ) ( ) ( ) 2 m = 1 min m 2 1 i<j m (ut i u j ) 2 [Lem. 3]

50 Proof of equality. λ 1 = = λ m = n m and ut i u j = m n n(m 1),

51 Proof of equality. λ 1 = = λ m = n m and ut i u j = i.e., {u 1,..., u m } is set of equiangular lines with ( ) m n (u i, u j ) = arccos. n(m 1). m n n(m 1),

52 m = n : u T i u j = 0

53 m = n : u T i u j = 0 m = n + 1 : u T i u j = 1 n

54 m = n : u T i u j = 0 m = n + 1 : u T i u j = 1 n m = ( ) n+1 2 : u T i u j = 1 n+2

55 m = n : u T i u j = 0 m = n + 1 : u T i u j = 1 n m = ( ) n+1 2 : u T i u j = 1 n+2 Lemmens-Seidel 73: n = 4 Not sharp :(

56 m = n : u T i u j = 0 m = n + 1 : u T i u j = 1 n m = ( ) n+1 2 : u T i u j = 1 n+2 Lemmens-Seidel 73: n = 4 Not sharp :( but for n = 7 and n = 23 is sharp too.

57 Corollary (Behrend 1937) Let K K 2 be in isodiametric position. Then vol(k) 3 D(K) 2 4. Equality holds iff K is an equilateral triangle.

58 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K.

59 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K. Then vol(k) vol(c)

60 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K. Then vol(k) vol(c) vol(conv([x 1, x 1 + u 1 ] [x 2, x 2 + u 2 ])) [Thm. 4]

61 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K. Then vol(k) vol(c) vol(conv([x 1, x 1 + u 1 ] [x 2, x 2 + u 2 ])) [Thm. 4] vol( 1 2 conv({±u 1, ±u 2 })) [Betke-Henk 93]

62 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K. Then vol(k) vol(c) vol(conv([x 1, x 1 + u 1 ] [x 2, x 2 + u 2 ])) [Thm. 4] vol( 1 2 conv({±u 1, ±u 2 })) = 1 1 (u1 T 2 u 2) 2 [Betke-Henk 93]

63 Proof. D(K) = 1, V = {u i } i [m], C := conv([x i, x i + u i ] : i [m]) K. Then vol(k) vol(c) vol(conv([x 1, x 1 + u 1 ] [x 2, x 2 + u 2 ])) [Thm. 4] vol( 1 2 conv({±u 1, ±u 2 })) = 1 1 (u1 T 2 u 2) 2 [Betke-Henk 93] /4 = 3 4 [Thm. 4]

64 Bibliography F. Behrend, Über einige Affinvarianten konvexer Bereiche, Math. Ann. 113 (1937), no. 1, B. González Merino, T. Jahn, A. Polyanskii, G. Wachsmuth, Hunting for reduced polytopes, Discrete Comput. Geom., P.W.H. Lemmens, J.J. Seidel, Equiangular lines, J. Algebra 24 (1973), J. Pál, Ein Minimal problem für Ovale, Math. Ann. 83 (1921),

65 Thank you for your attention!!

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE Abstract. We study the behavior of the so called successive inner and outer radii with respect to the p-sums