Rosenthal, A.; Szász, O., Eine Extremaleigenschaft der Kurven konstanter Breite, Deutsche Math.-Ver. 25, 278-282 (1916). ZBL46.1119.03.

The generalization to convex bodies in $\mathbf R^n$ reads $$\dfrac S{\omega_n}\leqslant\left(\dfrac D2\right)^{n-1}$$ where $D=$ diameter, $S=$ measure of boundary, $\omega_n=$ measure of the $(n-1)$-sphere $= \smash[b]{\dfrac{2\pi^{n/2}}{\Gamma(\frac n2)}}$: see (6) in §44 of

Bonnesen, T.; Fenchel, W., Theorie der konvexen Körper, Ergebnisse d. Math. 3, Nr. 1, VII + 164 p (1934). ZBL60.0673.01.

Bonnesen, T.; Fenchel, W., Theory of convex bodies, Providence, RI: AMS Chelsea Publishing. 164 S. (1971). ZBL0906.52001.

Bonnesen and Fenchel’s Theorie der konvexen Körper (1934, (6) p. 77) (translation) gives the generalization to a convex body $K\subset\mathbf R^n$ as $$ \mathrm{vol}(\partial K)\leqslant \omega_n\left(\dfrac{\mathrm{diameter}(K)}2\right)^{n-1} $$ where $\omega_n=\mathrm{vol}(S^{n-1})=2\pi^{n/2}\,/\,\Gamma(\frac n2)$. They seem to attribute it (p. 107) to Kubota (1925), with the planar case $n=2$ already in Blaschke (1915) and Rosenthal-Szász (1916).

Rosenthal, A.; Szász, O., Eine Extremaleigenschaft der Kurven konstanter Breite., Deutsche Math.-Ver. 25, 278-282 (1916). ZBL46.1119.03.

Rosenthal, A.; Szász, O., Eine Extremaleigenschaft der Kurven konstanter Breite, Deutsche Math.-Ver. 25, 278-282 (1916). ZBL46.1119.03.

The generalization to convex bodies in $\mathbf R^n$ reads $$\dfrac S{\omega_n}\leqslant\left(\dfrac D2\right)^{n-1}$$ where $D=$ diameter, $S=$ measure of boundary, $\omega_n=$ measure of the $(n-1)$-sphere $= \smash[b]{\dfrac{2\pi^{n/2}}{\Gamma(\frac n2)}}$: see (6) in §44 of

Bonnesen, T.; Fenchel, W., Theorie der konvexen Körper, Ergebnisse d. Math. 3, Nr. 1, VII + 164 p (1934). ZBL60.0673.01.

Bonnesen, T.; Fenchel, W., Theory of convex bodies, Providence, RI: AMS Chelsea Publishing. 164 S. (1971). ZBL0906.52001.