This is a consequence of the standard inequalities on the "mixed volumes" (or "quermaß integrals", the two are synonymous up to a constant) of convex sets: for every $i$, the $i$-th mixed volume $V_i(K)$ of a convex set is monotone in $K$, and for a $d$-dimensional convex set, $V_{d-1}(K)$ is twice the $(d-1)$-dimensional surface area of $K$ (while $V_d(K)$ is the $d$-dimensional volume; the quick definition in general is that $V_i(K)$ is the coefficient of $r^{d-i}$ in the polynomial giving the volume of the distance $r$ ball $K+B(0,r)$ around $K$, divided by the volume of $B(0,1)$ in dimension $d-i$).

See, for example, J. R. Sangwine-Yager's chapter "Mixed Volumes" (specifically §3) in the Handbook of Convex Geometry (vol. A), edited by Gruber & Wills (North-Holland 1993).

This is a consequence of the standard inequalities on the "intrinsic volumes" (or "quermaß integrals", the two are synonymous up to a constant) of convex sets: for every $i$, the $i$-th intrinsic volume $V_i(K)$ of a convex set is monotone in $K$, and for a $d$-dimensional convex set, $V_{d-1}(K)$ is twice the $(d-1)$-dimensional surface area of $K$ (while $V_d(K)$ is the $d$-dimensional volume; the quick definition in general is that $V_i(K)$ is the coefficient of $r^{d-i}$ in the polynomial giving the volume of the distance $r$ ball $K+B(0,r)$ around $K$, divided by the volume of $B(0,1)$ in dimension $d-i$).

See, for example, J. R. Sangwine-Yager's chapter "Mixed Volumes" (specifically §3) in the Handbook of Convex Geometry (vol. A), edited by Gruber & Wills (North-Holland 1993).

This is a consequence of the standard inequalities on the "mixed volumes" (or "quermaß integrals", the two are synonymous up to a constant) of convex sets: for every $i$, the $i$-th mixed volume $V_i(K)$ of a convex set is monotone in $K$, and for a $d$-dimensional convex set, $V_{d-1}(K)$ is twice the $(d-1)$-dimensional surface area of $K$.

See, for example, J. R. Sangwine-Yager's chapter "Mixed Volumes" (specifically §3) in the Handbook of Convex Geometry (vol. A), edited by Gruber & Wills (North-Holland 1993).

This is a consequence of the standard inequalities on the "mixed volumes" (or "quermaß integrals", the two are synonymous up to a constant) of convex sets: for every $i$, the $i$-th mixed volume $V_i(K)$ of a convex set is monotone in $K$, and for a $d$-dimensional convex set, $V_{d-1}(K)$ is twice the $(d-1)$-dimensional surface area of $K$ (while $V_d(K)$ is the $d$-dimensional volume; the quick definition in general is that $V_i(K)$ is the coefficient of $r^{d-i}$ in the polynomial giving the volume of the distance $r$ ball $K+B(0,r)$ around $K$, divided by the volume of $B(0,1)$ in dimension $d-i$).

See, for example, J. R. Sangwine-Yager's chapter "Mixed Volumes" (specifically §3) in the Handbook of Convex Geometry (vol. A), edited by Gruber & Wills (North-Holland 1993).