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For spanning slab normals, apply the maximum of finitely many Rademacher linear forms bound to $u_i=(V{\Lambda }^{1/2}{)}^T{a}_i$, whose squared norms are $a_i^{T}X{a}_i\le 1$. Some sign vector has scaling denominator squared at most $2\log (2m)$. Rademacher rounding for a semidefinite relaxation then produces a feasible vector of squared norm at least the displayed fraction of the SDP optimum. This is an existence guarantee from the probabilistic method.

Set $u_i=(V{\Lambda }^{1/2}{)}^T{a}_i$. SDP feasibility impliesChoose $\alpha =\sqrt{2\log (2m)}$. For the uniform sign vector, whose coordinates are independent Rademacher random variables, the supplied bound for the maximum of finitely many Rademacher linear forms givesA positive-probability event in this finite space contains at least one sign vector. ThusThe strict inequality in the supplied probability estimate matters: at the chosen threshold its right-hand side is zero.

For that sign vector, part (b) gives a feasible original point satisfying $\Vert x(\xi ){\Vert }_2^2\ge p_{\mathrm{SDP}}^{\ast }/[2\log (2m)]$. Taking the best original objective proves the unheaded concluding request as well:This is the logarithmic approximation bound for slab-constrained quadratic maximization, obtained by the probabilistic method. It is a finite-value rounding guarantee under the spanning/attainment conditions explained above.