Shape regularity controls affine simplex geometry

Use a singular-value decomposition $A_K=U\mathrm{\Sigma }{V}^T$. If $v_{\max }$ is a unit right singular vector for $s_{\max }=\mathrm{\parallel }{A}_K\mathrm{\parallel }$, the reference inscribed ball contains two points separated by $2r_{\widehat{S}}{v}_{\max }$; their images are separated by $2r_{\widehat{S}}{s}_{\max }$. Thus $s_{\max }\le h_K\mathrm{/}(2r_{\widehat{S}})$. For a unit right singular vector $v_{\min }$ and corresponding left vector $u_{\min }$, the width of $K$ in direction $u_{\min }$ equals $s_{\min }$ times the width of the reference simplex in direction $v_{\min }$, which is at most $s_{\min }{D}_{\widehat{S}}$. Every ball of radius $t<{\rho }_K$ contained in $K$ has width $2t$ in every direction, so letting $t\uparrow {\rho }_K$ gives $2{\rho }_K\le s_{\min }{D}_{\widehat{S}}$. Since $h_K\mathrm{/}{\rho }_K\le \sigma$, $s_{\min }\ge 2h_K\mathrm{/}(\sigma D_{\widehat{S}})$.

Every singular value lies between $c{h}_K$ and $C{h}_K$ by step 1.1. Their product is $\mathrm{\mid }\det A_K\mathrm{\mid }=\det A_K$ because the reference map is orientation preserving. Hence $c^{\mathrm{\prime }}{h}_K^d\le \det A_K\le C^{\mathrm{\prime }}{h}_K^d$.

For a reference facet normal $\widehat{n}$, [F2] gives its physical surface Jacobian $J_F=\det (A_K)\mathrm{\mid }{A}_K^{-T}\widehat{n}\mathrm{\mid }$. The singular values of $A_K$ from step 1.1 are between $c{h}_K$ and $C{h}_K$, so $\mathrm{\mid }{A}_K^{-T}\widehat{n}\mathrm{\mid }$ is between $c{h}_K^{-1}$ and $C{h}_K^{-1}$. Together with the determinant bounds this gives $c{h}_K^{d-1}\le J_F\le C{h}_K^{d-1}$. The simplex pyramid identity $\mathrm{\mid }K\mathrm{\mid }=\mathrm{\mid }F\mathrm{\mid }{H}_F\mathrm{/}d$ gives $H_F=d\mathrm{\mid }K\mathrm{\mid }\mathrm{/}\mathrm{\mid }F\mathrm{\mid }$, so step 2.1 and these facet bounds show $c{h}_K\le H_F\le C{h}_K$. This proves all claimed geometric scalings.