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Hedgehogs of constant width and equichordal points

100%

Martinez-Maure Y.

Annales Polonici Mathematici

1997

tom 67

nr 3

285-288

We give a characterization of convex hypersurfaces with an equichordal point in terms of hedgehogs of constant width.

On-line covering of the unit cube by boxes and by convex bodies

100%

Januszewski J. , Lassak M.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

tom Vol. 51, no 3

309-317

The paper presents a method of q-adic on-line covering of the unit d-dimensional cube I[sup]d by arbitrary sequence of boxes of side lengths of the form q[sup]-k for k [is an element of {O, l, 2,...} whose total volume is a number of the order of magnitude 2[sup]d. We also show that every sequence of boxes of side lengths at most 1 and of the total volume at least 4[sup]d. 2.566 ... permits an on-line covering of I[sup]d. Moreover, we estimate the total volume of sequences of convex bodies of diameters at most l which permit an on-line covering of I .

Some results about +nieskończoność-, -nieskończoność- and i-chord functions

86%

Soranzo A. , Sójka G.

Demonstratio Mathematica

2006

tom Vol. 39, nr 1

183-194

This paper concerns determination of a convex bodies by values of +oo- -oo-and i-chord functions. We prove that any at least two-dimensional convex body is not determined by values of -oo-chord functions at any two internal points. We also present some positive results on determination of convex bodies using -oo- or +oo-chord function at one point and i-chord function at other one.

Zagadnienie Auerbacha-Banacha-Mazura-Ulama pakowania worka ziemniaków

72%

Lassak M.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

1998

tom [Z] 34

49-59

Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

72%

Klee V. , Veselý L. , Zanco C.

Studia Mathematica

1996

tom 120

nr 3

191-204

For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class of all smooth bodies in X is stable with respect to both ∓ and γ̅. In our paper it is shown that when X is separable, these stability properties of rotundity (resp. smoothness) are actually equivalent to the reflexivity of X. The characterizations remain valid for each nonseparable X that contains a rotund (resp. smooth) body.

Banach-Mazur distance between convex quadrangles

72%

Lassak M.

Demonstratio Mathematica

2014

tom Vol. 47, nr 4

989--993

It is proved that the Banach–Mazur distance between arbitrary two convex quadrangles is at most 2. The distance equals 2 if and only if the pair of these quadrangles is a parallelogram and a triangle.