View
213
Download
0
Category
Tags:
Preview:
Citation preview
1.Introduction
Problem: to determine some inequalities describing geometric properties of the chords halving the area of a planar bounded convex set K.
- A. Ebbers-Baumann, A. Grüne, R. Klein: Geometric dilation of closed planar curves: New lower bounds. To appear in Theory and Applications dedicated to Euro-CG ’04, 2004.
2. Definitions
2.1 Halving partner.
Let K be a planar convex set.
Let p be a point on .
Then the unique halving partner p' on
is the intersection point between the straight line pp' halving the area of K and its boundary.
KK
2. Definitions
2.3 v-halving distance:
is the distance of the halving pair with direction v.
),(max:)(1
vKhKH ASv
A
),(min:)(1
vKhKh ASv
A
),( vKhA
Proof of Proposition 1:
1. it is trivial.
2.
Rotating v in there is at least, by continuity, a direction v0 such that the maximal chord in this direction divides K into two subsets of equal area. Then:
3. For every v, Then:
)()( KKHA )()( KHKD A
1S
)(),(),()( 00 KHvKhvKlK A
),(),( vKhvKl A
)(),(min),(min11
KhvKhvKl AASvSv
3. Overview of the results
ω D p r R A
AH none DH A pH A 21
none RH A 2 none
AH rH A 2
Ah Ah DhA phA 1
RhA 2
Ah none none none none
3. Overview of the results
ω D p r R A
AH none DH A pH A 21
none RH A 2 none
AH rH A 2
Ah Ah DhA phA 1
RhA 2
Ah none none none none
CS
CS ?, ?,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
Rh A 2
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
Rh A 2
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
pH A 43
rH A 2
RH A 433
AH A
42
Ah Ah Dh A ph A 1
rh A 3 Rh A 2
Ah A 32
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
Lemma 1 (Kubota):
If is a convex body, then
Lemma 2 (Grüne , Martínez, – – , Segura) :
If is a convex body, then
This bound cannot be improved.
2K
2DA
2K
DhA A
Proposition 2:
If is a convex body, then .
This bound cannot be improved.
Lemma 3:
If is a convex body, and is an arbitrary direction, then .
This bound cannot be improved.
2K 2Ah
2K 1Sv2lhA
Proof of the Lemma 3:
ppvhA )(
qqvl )(
21 llc
rightleft AA
)()( 22 vh
A
vl
AA
A
rightrightleft
21
)()(
2
2
rightright
right
rightleft
rightA
AAA
AAA
vlvh
Proof of Proposition 2:
Let be the direction such that
Then we get:
1Sv )(vhh AA
22
)()(
3
vlvhh
Lemma
AA
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
Rh A 2
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
Rh A 2
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
rh A 3
Rh A 2
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
rh A 3 Rh A 2 Ah A 32
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
rh A 3 Rh A 2 Ah A 32
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
pH A 43
rH A 2 AH A
42
Ah Ah Dh A ph A 1
rh A 3 Rh A 2 Ah A 32
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
3. Overview of the results
ω D p r R A
AH n o n e DH A pH A 21
n o n e RH A 2 n o n e
AH AH DH A 43
pH A 43
rH A 2 RH A 433
AH A 42
Ah Ah Dh A ph A 1
rh A 3 Rh A 2 Ah A 32
Ah 2
1Ah n o n e n o n e rh A 2 n o n e n o n e
CS
CS
?, ?,
EC ,
4. Conjecture and open problems4.1 In the family of all bounded convex sets
where the maximum is attained if and only if K is a disc. The conjecture was first posed by Santaló. The best bound known up to now, which is a consequence of Pal’s Theorem, is
...12838.12
)(
)(
KA
KhA
...31607.13)(
)( 4 KA
KhA
4. Conjecture and open problems4.2 Are discs the only planar convex sets with constant v-halving distance? Equivalently, is the lower bound of the ratio
attained ONLY by a disc?
1)()(
KhKH
A
A
5. Final remark
The chords halving the area of a planar bounded convex set are involved in the so called fencing problems which consider the best way to divide by a “fence” such sets into two subsets of equal area.
5. Final remark
The chords halving the area of a planar bounded convex set are involved in the so called fencing problems which consider the best way to divide by a “fence” such sets into two subsets of equal area.
- H. T. Croft, K. J. Falconer, R. K. Guy: Unsolved problems in Geometry. Springer-Verlag, New York (1991), A26;
- C.M, C. Peri, S. Segura Gomis: On fencing problems, J. Math. Anal. Appl. (2004), 464-476.
Recommended