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Search: MSC category 51K05 ( General theory )

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1. CMB Online first

Yumei, Ma
 Isometry on linear n-G-quasi normed spaces This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries on $n$-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces. Keywords:$n$-G-quasi norm, Mazur-Ulam theorem, Aleksandrov problem, $n$-isometry, $n$-0-distanceCategories:46B20, 46B04, 51K05

2. CMB 2009 (vol 52 pp. 407)

Lángi, Zsolt; Naszódi, Márton
 On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies The Bezdek--Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $\Re^d$ is $2^d$. Nasz\'odi proved that the quantity in question is not larger than $2^{d+1}$. We present an improvement to this result by proving the upper bound $3\cdot2^{d-1}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies. Keywords:Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodalityCategories:52C17, 51N20, 51K05, 52A21, 52A37
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