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Summary
Dedication to Kroly Bezdek
Periodica Mathematica Hungarica
Authors:
Károly Bezdek
and
Robert Connelly
Abstract
Let 0 < c < s be fixed real numbers such that
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$${c \mathord{\left/ {\vphantom {c s}} \right. \kern-\nulldelimiterspace} s} \leqslant {{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$
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, and let f : E2 → E
d
for d ≥ 2 be a function such that for every p, q ∈ E
2 if p − q = c, then f(p) − f(q) ≤ c, and if p − q = s, then f(p) − f(q) ≥ s. Then f is a congruence. This result depends on and expands a result of Rdo et. al. [9], where a similar result holds, but for
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$${{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 3}} \right. \kern-\nulldelimiterspace} 3}$$
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replacing
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$${{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$
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. We also present a further extensions where E2 is replaced by E
n
for n > 2 and where the range of c/s is enlarged.
Periodica Mathematica Hungarica
Authors:
Károly Bezdek
and
Robert Connelly
Abstract
Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in \documentclass{aastex}
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$$\mathbb{E}^d$$
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, which are centers of balls B
d
(p
i
, r
i
) and B
d
(q
i
, r
i
) of radius r
i
, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q,
then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension
of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections
of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted
to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way.
Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals
of functions (called flower weight functions) over \documentclass{aastex}
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, where the functions in question are constructed from maximum and minimum operations applied to functions each being radially
symmetric monotone decreasing and integrable.
Studia Scientiarum Mathematicarum Hungarica
Authors:
Robert Connelly
and
Balázs Csikós
A bicycle ( n , k )-gon is an equilateral n -gon whose k -diagonals are equal. S. Tabach-nikov proved that a regular n -gon is first-order flexible as a bicycle ( n , k )-gon if and only if there is an integer 2 ≦ r ≦ n -2 such that tan (π/ n ) tan ( kr π/ n ) = tan ( k π/ n ) tan ( r π/ n ). In the present paper, we solve this trigonometric diophantine equation. In particular, we describe the family of first order flexible regular bicycle polygons.
