Jensen-type geometric shapes



Shapes; Platonic shapes; sphere; ball; Jensen's inequality


We present both necessary and sufficient conditions for a convex closed shape such that for every convex function the average integral over the shape does not exceed the average integral over its boundary. It is proved that this inequality holds for n-dimensional parallelotopes, n-dimensional balls, and convex polytopes having the inscribed sphere (tangent to all its facets) with the centre in the centre of mass of its boundary.


de la Cal, Jesús and Javier Cárcamo. "Multidimensional Hermite-Hadamard inequalities and the convex order." J. Math. Anal. Appl. 324, no. 1 (2006): 248-261.

de la Cal, Jesús, Javier Cárcamo and Luis Escauriaza. "A general multidimensional Hermite-Hadamard type inequality." J. Math. Anal. Appl. 356, no. 2 (2009): 659-663.

Cover, Thomas M. and Joy A. Thomas. Elements of Information Theory 2nd ed. New York: Wiley-Interscience, 2006.

Dragomir, Sever Silvestru and Charles E. M. Pearce. Selected Topics on Hermite-Hadamard Inequalities. Victoria University: RGMIA Monographs, 2000.

Niculescu, Constantin P. "The Hermite-Hadamard inequality for convex functions of a vector variable." Math. Inequal. Appl. 5, no. 4 (2002): 619–623.

Niculescu, Constantin P. and Lars-Erik Persson. Convex Functions and their Applications. A Contemporary Approach, 2nd Ed. Vol. 23 of CMS Books in Mathematics. New York: Springer-Verlag, 2018.




How to Cite

Pasteczka, P. (2020). Jensen-type geometric shapes. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 19, 27–33. Retrieved from