Wikipedia:Reference desk/Archives/Mathematics/2017 March 11

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March 11

Containment

Is that true that whenever n-dimensional ellipsoid A contains a cube B, then A contains also the minimal (w.r.t containment) sphere C that contains B?

For example, if n-dimensional ellipsoid A contains $\left[-{\frac {1}{\sqrt {n}}},{\frac {1}{\sqrt {n}}}\right]^{n}$, than it contains the unit sphere. Is that true? 37.142.168.147 (talk) 08:11, 11 March 2017 (UTC)

Square in an ellipse, circle not so much...

Unless I misunderstand the question, no. See the sketch on the right for a counterexample of the 2d-case. --Stephan Schulz (talk) 12:32, 11 March 2017 (UTC)

Thank you! 31.154.81.69 (talk) 20:30, 11 March 2017 (UTC)

Quadratics

If the inequality $x^{T}Ax+bx+c\geq 0$ holds for all the vectors $\{e_{i}\}$, than it holds also for their span? 31.154.81.69 (talk) 20:34, 11 March 2017 (UTC)

@31.154.81.69: Your equation makes no sense, since the term bx is not a scalar when b is a mere scalar. If you mean for b to be instead a covector, then this is not true in general unless A is a positive-definite matrix.--Jasper Deng (talk) 05:53, 12 March 2017 (UTC)

Actually, it is not even enough that A be positive-definite. The whole quadratic form has to be positive definite, or the bx term can ruin things (consider the value on -ke_i for a large scalar k).--Jasper Deng (talk) 19:34, 12 March 2017 (UTC)

Yes, the statement is not true. It appears to be a homework problem, so I don't want to answer in detail, but 2 dimensional counterexamples are easy to come by. A hint: if $e_{i}$ is a solution, then $-e_{i}$ is in the span of the solutions (the counterexample I came up with uses a large b rather than a large k).--Wikimedes (talk) 03:36, 13 March 2017 (UTC)