Suppose that $\sum_{i=1}^{n}\lambda_{i}=1$, where $\lambda_{i}>0$, and $\sum_{i=1}^{n}x_{i}^{2}=1$, where $x_{i}>0$. Does one have $n^{3/2}\min_{1\le i\le n}\lambda_{i}x_{i}\le B$ for some constant $B$ (independent of $n$)? Thanks.
Is there a constant for this?
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real-analysis
analysis
inequality
multivariable-calculus
1 Answers
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The largest $\min_i \lambda_i x_i$ can be is when all $\lambda_i = 1/n$ and all $x_i = 1/\sqrt{n}$, which makes the left side $1$.
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0A continuous function on a compact set attains a maximum (to get a compact set, you have to change >0 to $\ge 0$, but the value is $0$ when some $\lambda_i$ or $x_i$ is $0$). – 2012-12-07