The lattice point propery of the ellipse

Let L2 be an orthogonal lattice in a plane q and F a figure in q. We define that F has the l.p.c.p.(lattice point covering property) if for every position of F in the plane, the figure F includes at least one lattice point.

It is very simple to see that the circle c0 of diameter 2R=21/2 has the l.p.c.p. So every figure including c0 has the l.p.c.p. For example the equilateral triangle including c0 has side equal to 2.449 and obviously has the l.p.c.p. Nevertheless we can prove that an equilateral triangle with side 2.154 has l.p.c.p. That is, the problem is to find the minimum conditions between the elements of the figure F, so that the figure F to be able for the l.p.c.p.

Another example for L3. A cube of side 31/2 has obviously the l.p.c.p. because includes a sphere of diameter 31/2. But it has been proved by M. Henk that a cube of side  21/2 has the l.p.c.p. Remains open the problem for n-cube. M. Henk and the author solved the problem for the ellipsoid in Ln and here is the solution for l.p.c.p. of the ellipse in the plane.

The Erdos-Mordell Inequality

Let ABC be a triangle and an interior point M. We denote by AM =x1, BM = x2, CM = x3. The distances of the point M from BC, CA, AB are denoted by p1, p2, p3. Erdos-Mordell inequality asserts:

x1 + x2 + x3 no less than 2[p1 + p2 + p3]

The above problem proposed by P.Erdos in the American Mathematical Monthly in 1935 and solved byI.J.Mordell and D.F.Borrow in 1937. Later many Mathematicians obtained solutions and for a long time the problem was in the air.

For some more solutions you can download my paper on the Erdos-Mordell Inequality.

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